Its pretty difficult to diagnose over the internet since there is so much involved. Heck, your GP won't even do it in person (but I understand why). But some thoughts.
A child with ADHD can hyperfocus or what you call obsession. Also kids of this age aren't very good at eye contact anyway. Because of that and a few other things you have said, I tend to doubt that autism is the problem. He certainly has many of the symptoms of ADHD.
And you mentioned one or two things that also make me think of SIDS or sensory integration disorder. There is a really good website called simply "sensory processing disorder" or "spd". Google that to learn more about what areas it can affect a child. And if you check out this post, scroll down to the last one by specialmom - you can learn even more http://www.medhelp.org/posts/Child-Behavior/SIDS-ADHD/show/1575545#post_7156293
But, as I said, he certainly has many of the symptoms of ADHD. Probably the most complete information I have found for the symptoms etc., is here
It will probably give you more information then you want to know.
Finally, just some thoughts on discipline. JGF25 had some very good points which I certainly agree with. But first and foremost - at this age there is nothing that you can do to punish him for what he does at school ...because it won't work. Maybe when he is 10, certainly not 5. At this age any behavioral change modification must be immediate and consistent! To do something to him for something he did hours earlier is a waste of time because he does not have the capability to link these things together.
What you can do at home is to work on behaviors that will carry over to school. There are several great series of books that are meant to be read to 4 to 7 year olds. One example is here and you will find more listed below on this site. Here are two sample choices "http://www.amazon.com/When-Feel-Angry-Way-Books/dp/0807588970/ref=pd_sim_b_2
A book that might help you is one we recommend alot on the child behavior forum and its Oh, I would also suggest getting the book, "SOS Help for Parents," by Lynn Clark. It gives a very detailed system using the timeout method that will work. It does sound like you are using pretty good techniques to change behavior at home. But you need to remember that you are changing behavior and it doesn't happen overnight. It takes at least 3 weeks on consistent, immediate reinforcement, and thats for normal kids. As JGF25 pointed out some kids just don't have the ability to change. Well, they do but it takes a lot longer and a lot more work.
A final book which I recommend a lot is "The ADD/ ADHD Answer book," by Susan Ashley. If it does look like he has ADHD, then this is certainly a book that you are going to need. Hope, I haven't given you too much info here. Please post if you need any clarification or need more info. Best wishes.