Trinity emerges from Bhrahmn (different sects call it differently such as Kala, SadaShiva, AdiNarayana, AdiShakti etc.). This is discussed in this post:
What is the Life span of Hindu gods
Who is the mother of Lord Shiva?
As discussed above, Brahman is mother/father of Lord Shiva.
From various Purana:
The Brahman is the Lord who is the origin of everything. It has neither form nor is it without form. The Brahman has no beginning; it is independent of all action. The Brahman is huge; it is everywhere. It has neither name nor is it without name. It has neither name nor is it without name. The Brahman not only has no form, it is beyond all form. It had no origin, no beginning or no end.
Most of the stories which tell how Trinity (Trimurti i.e Brahma, Vishnu and Shiva) came into existence are not from the very first Kalpa and that's how confusion occurs.
"Who is your father, Lord?". Lord Shiva replied that Brahma is his father. So who is the mother?
This is not Shiva but Rudra. Many scholars use Rudra and Shiva interchangeably, another reason for the confusion.
Rudra was born from Lord Brahma's teardrops. Hence, no one was Rudra's mother. ~Vayu Purana
“Who is your grandfather?” .Then the saint went on to ask who is your grandfather. Lord Shiva replied that Vishnu is my grandfather.
Well, as described above Lord Brahma is Rudra's father. Now how that happened is as follows:
At the end of the last kalpa, there was a minor destruction and the world was flooded with water. Vishnu alone slept on this water, balanced on the hood of the great snake, Ananta. While Vishnu was thus sleeping, a lotus sprouted from his navel. It was a huge lotus and extended for a hundred yojanas. And it shone with radiance. Vishnu began to play with the lotus. While Vishnu was thus playing, Brahma arrived. "Who are you and why are you sleeping on this water?" asked Brahma. "I am Vishnu and I am the lord of everything." replied Vishnu. "But who are you and where are you going?" "How can you be the lord of everything?" asked Brahma. "I am the lord of everything that is in the universe. Everything that will be there in the universe is already there inside my stomach. If you don't believe me, why don't you enter my stomach and see for yourself?" Vishnu's curiosity was stirred and he entered Brahma's stomach. Inside the stomach he was greatly surprised to find all the worlds that would be there in the universe. The mountains and the oceans were all there. So were all the living beings who would be created. Visnu spent one thousand years inside Brahma's stomach, marvelling at these wonders. But he could find neither the end nor the beginning of Brahma's stomach. He finally made his exit through Brahma's mouth. Vishnu told Brahma, "I bow down before you, you re indeed the lord of everything. There are many marvels that I saw inside your stomach, many are the worlds that are there. But I can also rival you. Why don"t you enter my stomach? I too can show you many worlds there." Brahma entered Vishnu's stomach and saw many worlds there, as Vishnu had promised. Brahma spent several years inside the stomach, but could find neither its end nor its beginning. Meanwhile, Vishnu had closed all the exits from his body and Brahma could find no way of coming out. He finally made his body very small and came out through Vishnu"s navel. He clambered up the stalk of the lotus and seated himself on the lotus. Since a lotus is called padma and yoni means birth-place, Brahma came to be known as Padmayoni.
While all this was going on, Shiva arrived on the scene. He held a trident in his hand. Such was the speed of Shiva"s arrival that huge tidal waves were created in the water. Strong winds started to blow. "What is all this? Why are you shaking the lotus so and creating tidal waves?" Brahma asked Vishnu. "Who speaks from my navel?" said Vishnu. "Don't you remember?" replied Brahma. "I am Brahama. You had entered my stomach and, thereafter I had entered your stomach. But you had closed all the exits, so that I had no way of getting out. In fact, I ought to be downright angry with you for treating me thus. I had to emerge through your navel and am now seated on the lotus." "I am sorry," said Vishnu. "I had not meant to insult you. I had only wished to play with you for a while. Let us now be friends. And as a token of your friendship, please grant me the boon that you will henceforth be known as my son." "I agree," replied Brahma. "At first, I thought that you were shaking the lotus and creating these tidal waves in the water. But now I see that there is another creature who is advancing towards us. His visage is terrible. He has ten arms and he holds a trident. Who is this demon?" Vishnu told Brahma that this was none other than Shiva, the destroyer, and that the two of them should pray to Shiva. But Brahma flatly refused to do this. He would not acknowledge Shiva as a superior. Vishnu's persuasion however bore fruit and both Brahma and Vishnu started to pray to Shiva. "What can I do for you?" asked Shiva. "What boon do you desire?" "I am sorry that I doubted you," replied Brahma. "Please forgive me. And as a token of your pardon, please grant me the boon that you will be born as my son." Shiva gladly granted this boon. ~Vayu Purana
This should be noted that this all happened in some Kalpa and not in the beginning of the very first Kalpa. As 360*50 Kalpas have been passed in current Brahma's lifespan so this nowhere prove that Lord Vishnu created Lord Brahma. At the time of this story, Lord Brahma was there. But as described in this story Lord Vishnu can be considered Lord Brahma's father.
“Who is your great-grandfather?” The saint further queries as to who is the great-grandfather. Lord Shiva stuns him by saying that I am my great-grandfather.
This is SadaShiva (Bhrahmn in Shaivism) who creates Brahma, Vishnu and Shiva. In Vaishnavism, same Bhrahmn is called as AdiNarayana.
"Lord Shiva emerged from the pillar" is it real?
This is partially right. When fight between Lord Vishnu and Lord Brahma was going on, Lord Shiva came to know about this and emerged from the pillar. That means Shiva was there even before he emerges from the pillar.