Did Lord Buddha really not believe in God?
First of all among the Six Schools of Hinduism the schools like Vaisheshika founded by Maharshi Kanada, Samkhya founded by Maharshi Kapila etc.. do not believe in God. They are non-theistic schools which do not believe in God, but believe that Vedas are authority. So, there are itself schools in Hinduism which do not believe in God.
Regarding Buddha it depends on what one mean by God? If you call God as eternal, omnipotent everwitness entity then Buddhism certainly doesn't speculate on these things. If you call God as higher being, living in heavens, having supernatural powers etc.. then Gautama Buddha certainly talked of these type of Gods.
For eg. In the Brahmajala Sutra of Pali canon (note that Pali canons are considered the most authentic part of Buddhist scriptures; actually Theravada schools holds only Pali canon as authentic) Buddha states:
There comes a time, bhikkhus, when after the lapse of a long period this world contracts (disintegrates). While the world is contracting, beings for the most part are reborn in the Ābhassara Brahma-world. There they dwell, mind-made, feeding on rapture, self-luminous, moving through the air, abiding in glory. And they continue thus for a long, long period of time.
40. "But sooner or later, bhikkhus, after the lapse of a long period, there comes a time when this world begins to expand once again. While the world is expanding, an empty palace of Brahmā appears. Then a certain being, due to the exhaustion of his life-span or the exhaustion of his merit, passes away from the Ābhassara plane and re-arises in the empty palace of Brahmā. There he dwells, mind made, feeding on rapture, self-luminous, moving through the air, abiding in glory. And he continues thus for a long, long period of time.
Then Buddha goes on stating:
"Thereupon the being who re-arose there first thinks to himself: 'I am Brahmā, the Great Brahmā, the Vanquisher, the Unvanquished, the Universal Seer, the Wielder of Power, the Lord, the Maker and Creator, the Supreme Being, the Ordainer, the Almighty, the Father of all that are and are to be. And these beings have been created by me. What is the reason? Because first I made the wish: "Oh, that other beings might come to this place!" And after I made this resolution, now these beings have come.'
"And the beings who re-arose there after him also think: 'This must be Brahmā, the Great Brahmā, the Vanquisher, the Unvanquished, the Universal Seer, the Wielder of Power, the Lord, the Maker and Creator, the Supreme Being, the Ordainer, the Almighty, the Father of all that are and are to be. And we have been created by him. What is the reason? Because we see that he was here first, and we appeared here after him.'
43. "Herein, bhikkhus, the being who re-arose there first possesses longer life, greater beauty, and greater authority than the beings who re-arose there after him.
So, it's clear from the above passage that Buddha talked of powerful deities like Brahma. But he is not eternal God. He is just like Devata of Hinduism (ie. Devas having lifespan).
What does Buddha speak of all Eternal and omnipotent God?
Actually speculating these type of questions are not considered useful in Buddhism. And the Buddhist text hold that one that is ultimately responsible for his Nirvana is himself (and not any other Gods). For instance if someone asks this question to Buddha, he would have replied as:
Person: Does the Eternal, Omnipotent God exist?
Buddha:Is knowing whether the Eternal God exists or not useful for the cessation of sufferings?
Similar question were asked to Buddha by Malyunkaputta in the Malunkaputta Sutta like whether cosmos is finite or infinite, whether body and soul are same or body and soul are different etc.. and Buddha replies as:
"It's not the case that when there is the view, 'After death a Tathagata exists,' there is the living of the holy life. And it's not the case that when there is the view, 'After death a Tathagata does not exist,' there is the living of the holy life. And it's not the case that when there is the view, 'After death a Tathagata both exists & does not exist,' there is the living of the holy life. And it's not the case that when there is the view, 'After death a Tathagata neither exists nor does not exist' there is the living of the holy life. When there is the view, 'After death a Tathagata exists'... 'After death a Tathagata does not exist'... 'After death a Tathagata both exists & does not exist'... 'After death a Tathagata neither exists nor does not exist,' there is still the birth, there is the aging, there is the death, there is the sorrow, lamentation, pain, despair, & distress whose destruction I make known right in the here & now. "
So, whether you know or do not know these things there is still ageing, still death still suffering etc.. So, Buddha focuses in removal of suffering and declared such things as Undeclared:
"So, Malunkyaputta, remember what is undeclared by me as undeclared, and what is declared by me as declared. And what is undeclared by me? 'The cosmos is eternal,' is undeclared by me. 'The cosmos is not eternal,' is undeclared by me. 'The cosmos is finite'... 'The cosmos is infinite'... 'The soul & the body are the same is undeclared by me'... 'The soul is one thing and the body another is undeclared by me'... 'After death a Tathagata exists'... 'After death a Tathagata does not exist'... 'After death a Tathagata both exists & does not exist'... 'After death a Tathagata neither exists nor does not exist,' is undeclared by me.
Buddha also makes clear why he undeclared such things:
"And why are they undeclared by me? Because they are not connected with the goal, are not fundamental to the holy life. They do not lead to disenchantment, dispassion, cessation, calming, direct knowledge, self-awakening, Unbinding. That's why they are undeclared by me.
"And what is declared by me? 'This is stress,' is declared by me. 'This is the origination of stress,' is declared by me. 'This is the cessation of stress,' is declared by me. 'This is the path of practice leading to the cessation of stress,' is declared by me. And why are they declared by me? Because they are connected with the goal, are fundamental to the holy life. They lead to disenchantment, dispassion, cessation, calming, direct knowledge, self-awakening, Unbinding. That's why they are declared by me.
Buddha also gives the parable of poisoned arrow to those who try to speculate such things:
"It's just as if a man were wounded with an arrow thickly smeared with poison. His friends & companions, kinsmen & relatives would provide him with a surgeon, and the man would say, 'I won't have this arrow removed until I know whether the man who wounded me was a noble warrior, a brahman, a merchant, or a worker.' He would say, 'I won't have this arrow removed until I know the given name & clan name of the man who wounded me... until I know whether he was tall, medium, or short... until I know whether he was dark, ruddy-brown, or golden-colored... until I know his home village, town, or city... until I know whether the bow with which I was wounded was a long bow or a crossbow... ..... ..... ...... ...... ....... ....... ........ ........ ...... He would say, 'I won't have this arrow removed until I know whether the shaft with which I was wounded was that of a common arrow, a curved arrow, a barbed, a calf-toothed, or an oleander arrow.' The man would die and those things would still remain unknown to him.
So, such Metaphysical speculation isn't considered useful in Buddhism, they are only seen by direct experience. So, these type of topics are like Mute subjects in Buddhism.
"Did Lord Buddha have any oppositions with the Vedas?"
Buddha gives a long discussion and reasoning to explain how knowledge of only 3 Vedas is not sufficient. It comes in Tevigga Sutta.
"with the Brahmins?"
Buddha says that Brahmanas of previous times were following a good religion. In the "Brahmana Dhammika Sutta" (II,7) of the Suttanipata Buddha speaks:
The elderly Brahmins asked him, "Do the present Brahmans follow the same rules, practice the same rites, as those in the more ancient times?"
The Buddha replied, "No."
The elderly Brahmins asked the Buddha that if it were not inconvenient for him, that he would tell them of the Brahmana Dharma of the previous generation.
The Buddha replied: "There were formerly rishis, men who had subdued all passion by the keeping of the sila precepts and the leading of a pure life...Their riches and possessions consisted in the study of the Veda and their treasure was a life free from all evil...The Brahmans, for a time, continued to do right and received in alms rice, seats, clothes, and oil, though they did not ask for them. The animals that were given they did not kill; but they procured useful medicaments from the cows, regarding them as friends and relatives, whose products give strength, beauty and health."
"Vedic rituals, cultures?"
Buddha certainly opposed Vedic rituals such as Animal sacrifice and focusing only on Karma kandas, etc.. However Buddha does pays respect to Vedas. As I discuss in my question here :
In the Buddhist Vinaya Pitaka of the Mahavagga (I.245) section the Buddha names these rishis. The names of the Vedic rishis were "Atthako, Vâmako, Vâmadevo, Vessâmitto, Yamataggi, Angiraso, Bhâradvâjo, Vâsettho, Kassapo, and Bhagu" but that it was altered by a few Brahmins who introduced animal sacrifices. The Vinaya Pitaka's section Anguttara Nikaya: Panchaka Nipata says that it was on this alteration of the true Veda that the Buddha refused to pay respect to the Vedas of his time.
So, Buddha refused to pay respect to Vedas of his time because he considered they were altered. Also in Sutta Nipata 192 Buddha states:
Vidwa Cha Vedehi Samechcha Dhammam Na Uchchavacham Gachhati Bhooripanjo.
People allow sense-organs to dominate and keep shuffling between high and low positions. But the scholar who understands Vedas understands Dharma and does not waver.
However here Veda can also mean knowledge. In the Sutta Nipata 846 Buddha states:
Na Vedagu Diththia Na Mutiya Sa Manameti Nahi Tanmayoso….
One who knows Vedas does not acquire false ego. He is not affected by hearsay and delusions.
As a Sidenote I want to add here two things:
Although there are procedures for animal sacrifice in Vedas but still ancient Rishis were also confused/speculated whether Vedas actually mean animal sacrifice or not. For eg in this chapter of Shanti Parva of Mahabharata in the Narayaniya section, Rishis say:
The Rishis said, The Vedic Sruti declares that in sacrifices the offerings should consist of (vegetable) seeds. Seeds are called Ajas. It behoveth you not to slay goats. Ye deities, that cannot be the religion of good and righteous people in which slaughter of animals is laid down. This, again, is the Krita age. How can animals be slaughtered in this epoch of righteousness?'
"Bhishma continued, While this discourse was going between the Rishis and the deities, that foremost of kings, viz., Vasu, was seen to come that way. Endued with great prosperity, the king was coming through the welkin, accompanied by his troops and vehicles and animals. Beholding king Vasu coming to that spot through the skies, the Brahmanas addressing the deities, said,--This one will remove our doubts. He performs sacrifices. He is liberal in making gifts. He always seeks the good of all creatures. How, indeed, will the great Vasu, speak otherwise,--Having thus spoken unto each other, the deities and the Rishis quickly approached king Vasu and questioned him, saying,--O king, with what should one perform sacrifices? Should one sacrifice with the goat or with herbs and plants? Do thou dispel this doubt of ours. We constitute thee our judge in this matter.--Thus addressed by them, Vasu joined his hands in humility and said unto them.--Tell me truly, ye foremost of Brahmanas, what opinion is entertained by you in this matter?
"'The Rishis said, The opinion entertained by us, O king, is that sacrifices should be performed with grain. The deities, however, maintain that sacrifices should be performed with animals. Do thou judge between us and tell us which of these opinions is correct.'
"Bhishma continued, 'Learning what the opinion was that was entertained by the deities, Vasu, moved by partiality for them, said that sacrifices should be performed with animals. At this answer, all the Rishis, endued with the splendour of the Sun, became very angry. Addressing Vasu who was seated on his car and who had (wrongly) taken up the side of the deities, they said unto him,--Since thou hast (wrongly) taken up the side of the deities, do thou fall down from heaven. From this day, O monarch, thou shalt lose the power of journeying through the sky. Through our course, thou shalt sink deep below the surface of the Earth.
Buddha being incarnation of Lord Vishnu is not present in any Vedas, neither it is in ValmikiRamayana, nor it is in the Vyasas Mahabharata. It is only present in some Puranas. Actually Harivamsha Purana; which is an appendix to Mahabharata doesn't lists Buddha as incarnation while it lists all major incarnations in Vishnu Parva. So, the thing that (Gautama) Buddha is an incarnation of Vishnu can also be only latter additions in some Puranas.