Ania, I can relate to so much of what you write about your son. The lack of self-motivation sure is frustrating.
My son is younger than yours (will be 8 next month) and exhibits many of the same traits. He would definitely play on the computer all day long if he could (but isn't that true of many adults, too!). He is impatient about doing homework, about practicing/studying for anything, or doing anything he views as unnecessary busywork. For example, he hates having to write an answer in a complete sentence instead of a short phrase. He wants to rush through schoolwork and get it over with, which means he makes careless mistakes. He often misses points because he doesn't read the complete directions. And yet, he cares very much about getting good grades and being viewed as smart. He is upset if he doesn't get good grades.
My son tends to get along well with either much older kids (if they will deign to play with him) or younger kids, because then he can be the leader. I also liked what you said about your son in this regard--that perhaps the younger kids don't question his strange play ideas.
I think your theory (too much freedom at school) is good. Is his computer time pretty much a free-for-all where he can do whatever he wants? Maybe you could work with the teacher to find ways to narrow the scope of what he can do with that time--create specific assignments he has to do on the computer.
