You could say that ‘he adapted very well to be at at a new school’ but it gives the sentence a very different meaning. ‘Adapted to be’ is giving us the reason why he adapted - he adapted because he wanted to be in a new school. ‘Adapted to being’ is telling us that he has already had experience at this new school and that he is functioning in a way that works for the school. In the second sentence we could use the gerund ‘using’ but we would have to change the sentence a bit and it would change the meaning completely. We could say ‘the car was adapted for using too much fuel’ - meaning someone changed the car to use less fuel because before it was using too much fuel. But when we say ‘the car was adapted for use as a taxi’ it means someone changed the car to function like a taxi. We are comparing the taxi with the car.

in the second sentence, the word ( use ) here is a noun.

in the first,   adapted to + gerund  it can be used when "adapted to" means "appropriate for/ suitable to.

Your grammar book should have a discussion of gerunds. If you don't have a grammar book, English Grammar in Use 4th edition by Raymond Murphy is excellent. The paper edition is reasonably priced and a PDF is available on the net.