Did Lord Rama really use the Brahmastra on Ravana to kill him?
Yes, he did. He used the Brahmastra given by sage Agastya. He used Brahmastra by the advice of Matali.
Then, the valiant Rama, who was reminded thus by Matali, took hold of a blazing arrow, which was given by Brahma and which in turn was given to him by the glorious sage, Agastya earlier in the battle-field and which looked like a hissing serpent.[6-108-3]
Above shloka shows that Rama used a Brahmastra against Ravana.
It would have even created a major natural disaster in the sky and on land as Brahmastra's power is just too great.
Yes, some disasters did happen when Rama aimed that Brahmastra.
While that excellent arrow was being fixed by Rama, all the beings were frightened and the earth trembled. [6-108-15]
That arrow, which was capable of causing death to the body, after taking away the life of Ravana and having been anointed with blood, penetrated the earth [6-108-19]
Lakshmana and the entire army was safe too and Lord Rama's chariot was not scratched a tiny bit.
Yes, they were safe because Rama only used it on Ravana and not others. Brahmastra is a divine astra which is invocated through mantras. It only destroys the enemy and bestows joy to the person who invokes it and the astra behaves as per the will of the person who aimed it. This was the reason Sri Rama controlled the power of the Brahmastra which was fired at Jayanta. It diluted the power to destroy an eye of Jayanta instead of his life (that was also a Brahmastra but it is Aishiika missile a grass blade missile. One needs to have the knowledge of the Aishiika missile to invoke divyastras into a grass blade. But this time against Ravana, a rod given by Brahma is used.)
That arrow bestowed joy on the monkey-leaders and destroyed the demons. It was made speedy, by tying various kinds of beautiful coloured feathers of Garuda, the king of eagles, to it. [6-108-12]
Making it sacred by a special formula as per the procedure specified in scriptures, Rama who was endowed with an extraordinary strength then fixed that arrow which was the foremost among the three worlds, capable of removing the fear of Ikshwaku dynasty, taking away the glory of the enemies and bestowing joy to one's own self on his bow.[6-108-13]
Sri Rama didn't want Lakshmana and others to be harmed so the missile obeyed the order by Rama.
But Ravana's body was still there.
Yes, Ravana's body was still there. Vibhishana performed the funeral rites to his brother after Sri Rama convinced him.
But Mahabharata says this differently. Sage Markandeya narrates the story of Ramayana to Yudhishtra during the exile. This occurs in the Mahabharata, Vana Parva Chapter 288
Then the descendant of Kakutstha, taking out of his quiver an excellent arrow furnished with handsome wings and golden feathers and a bright and beautiful head, fixed it on the bow with Brahmastra mantra. And beholding that excellent arrow transformed by Rama, with proper mantras into a Brahma weapon, the celestials and the Gandharvas with Indra at their head, began to rejoice. And the gods and the Danavas and the Kinnaras were led by the display of that Brahma weapon to regard the life of their Rakshasa foe almost closed. Then Rama shot that terrible weapon of unrivalled energy, destined to compass Ravana's death, and resembling the curse of a Brahmana on the point of utterance. And as soon, O Bharata, as that arrow was shot by Rama from his bow drawn to a circle, the Rakshasa king with his chariot and charioteer and horses blazed up, surrounded on all sides by a terrific fire. And beholding Ravana slain by Rama of famous achievements, the celestials, with the Gandharvas and the Charanas, rejoiced exceedingly. And deprived of universal dominion by the energy of the Brahma weapon, the five elements forsook the illustrious Ravana. And were consumed by the Brahma weapon, the physical ingredients of Ravana's body. His flesh and blood were all reduced to nothingness,--so that the ashes even could not be seen.
According to the narration of Sage Markandeya in the Mahabharata, there was no physical ingredient left of Ravana.
If Lord Rama used the Brahmastra then Ravana would have been fully aware of it then why didn't Ravana launch his own Brahmastra to counter it?
I think the answer lies in the question itself. Because we are not sure that Ravana had the Brahmastra. It is not mentioned in the Valmiki Ramyana or any other puranas that he used the Brahmastra prior to the war with Rama. We can only speculate. Even if possessed, to invoke a Brahmastra, there is a process. It would be not possible to invoke a Brahmastra if he doesn't remember the proper mantras (remember the case with Karna. At a point of time, he couldn't invoke a Brahmastra but later after recovering, he used it against Arjuna). This could be the possible reason that Ravana didn't use Brahmastra if he possessed the weapon.