Concerning Finny's relief when he gets confirmation that Gene is still his roommate, both of the above editors do a great job of answering the question. I'll just add a slightly different take on it.
When I read this section of the book, I see Finny looking for confirmation that he still has a "hold" on Gene, that he is still able to influence him. Finny thrives on getting Gene to do things that he wants Gene to do. Notice that Finny assumes Gene held the room for him, that Gene wouldn't let "them" put anyone else in the room. Of course, readers know this isn't the case, that Gene didn't have anything to do with it. Finny is elated because he still has this hold on Gene, even though physically he is crippled.
This is one more illusion on Finny's part.
Although we see Gene seem dependent on Finny's approval throughout the beginning of the book, the one who every now and then audibly expresses his feelings about the other is Finny.
In the beginning of the novel, Finny actually admits Gene is his best friend. Guys just don't say emotionally complimentary things to each other.
Finny needs a good friend, especially under the circumstances of his injury. He likes to pretend that not much is wrong, but if there is someone he can be a little bit real with, it's Gene. He's not completely transparent with Gene, but he is dependent on Gene for approval.
Even though Finny seems extremely confident and even though he had a dramatic reaction to Gene's visit to his house (after his injury), he still wants and needs Gene to like him. He still thinks of Gene as his friend, and if Gene continues to be his roommate, it helps Finny convince himself that the tree incident was an accident.
Finny's desire to be roommates with Gene also illustrates the novel's theme of inner peace. Finny does not want to be at "war" with any of the other boys. Throughout the novel--even after his injury--he develops activities which are intended to bring the boys together. Similarly, Finny would not be able to deal with a Devon School where his best friend was his rival or enemy.