The palace was very big <strike>that</strike> and gave a feeling of coldness. There were not a lot of pieces of furniture<strike>s</strike> in there, and he thought that the room in which he <strike>was</strike> stood (there are a lot of different words you could use here) was too big for the little things inside.

He thought that for nothing in the world he would exchange (trade might also be a good word here, instead of change) his little house for this palace.

Keep up the good work.

"He didn't want he wake up and think he'd been deserted." sounds a bit awkward in English. People will still understand you, but it would make more sense to say something like: "He didn't want him to wake up and think he'd been deserted." It's a small difference, but it makes more sense. I don't know the real reason, but the first one sounds a bit unnatural. I looked it up online and he is for when you're talking about the subject of a sentence, and him is used for the object.