A book in your room
Both of these sentences are correct English. Given what you said you wanted to express, sentence A is the better one (it says that you posess a book, and the book is in your room):
I have a book in my room.
Sentence B is also correct, but only says that the book is in your room, it does not say anything about whether the book is yours or not.
Living in your heart
Sentence A is actually fairly reasonable for this situation. To say you have somebody in your heart implies that you love them, and also that they are living (dwelling) there in your heart. You don't necessarily need to say anything more than that. In fact, you could even say less:
The girl is in my heart.
Sentence B is awkward. It could technically be grammatically correct, but only if interpreted in a way that you probably didn't mean (what it says is basically the same as "I make the girl live in my heart"). You could instead say (as you basically did to begin with):
I have the girl living in my heart
...which is more correct, and means that you have a condition, and the condition is that the girl is living in your heart.
(While "I have her in my heart" or "She is in my heart" probably imply that she's there because you want her to be in your heart, saying "I have her living in my heart" might actually imply that you have less choice in the matter (she would be living in your heart whether you wanted her to or not).)
Which is what I hope
There are actually a couple of problems with this. The first is that you can't really "hope" just by itself. You need to "hope for" something or "hope that" something happens, so you almost always need a preposition following that verb.
The other issue here is that you generally only hope for things which haven't already happened yet. Since you are saying that the girl is already in your heart, then it's not really something you hope for anymore. What you probably want to say in this case is that you would hope for it (if it weren't already true):
The girl is living in my heart, which is what I would hope for.
Or possibly, if you wanted to express that you wouldn't want to hope for anything more than that (or you don't believe if you did that it would come true):
The girl is living in my heart, which is all I could hope for.
(You could also use "want" instead of "hope for". In the first sentence it makes it a little softer. In the second sentence it makes it more clear that you have everything you want (whereas "all I could hope for" could be taken to mean you might want more, but just don't think it would ever happen).
Leaving out "what"
Instead of "which is what I would hope for", you could technically say "which I would hope for" instead (leaving out the "is what"), but it does change the feeling of the sentence, and would probably end up being awkward to most readers. The main problem is that the phrase "I would hope" is usually used in that way when somebody doesn't know whether something is true, but if it were true, they would hope something else to be true as well:
I really like mustard, which I would hope that they provide if they're going to serve us hamburgers.
Given this, using "which I would hope for" in this context would usually cause a reader to ask "if what happens?"