a. I bought an and the book.
I think you meant to say I bought a and the book, distinguishing between two separate books. But both I bought an and the book and I bought a and the book are incorrect. You cannot use a conjunction between articles of two separate nouns.
b. I want this and that book.
I think you can say this, yes, but it is very awkward. It would be better to say I want this book and that book.
c. This is his and her room.
Yes, this is OK, technically. If there is a girl and a boy who share a bedroom, you can say this. It would be more common to say, simply, This is their room.