Random list of thoughts, very poorly organized:
Is he in public school and has he had any kind of evaluation? What does his pediatrician say? If he can have an evaluation and get some kind of plan in place, you can make one of the "rules" that the teacher reports about his behavior every day - this relieves him of the need to lie and you of having to depend on his iffy interpretation of the day. My daughter is six. She comes home with a folder every day. There's a calendar stapled to one side. Her "color" is marked every day. I'm told if there are a lot of yellows or a red, there will be a note as to why so you don't have to ask and the kid doesn't feel backed into a corner to lie. They have the same system through third grade, I think. It helps a lot. I know what happened and I can send a note back in the folder to get more info if I need it (or I can email, her teacher is fantastic about responding to email).
As for keeping up with his stuff, does all his stuff have a place? We have found things much easier now that my daughter has an "after school" spot - coat, shoes, school bag, lunch box, library books all go in the same place. It's right inside the door. We don't have to hunt things because those things live there. Same with clothes - there's a hamper in her room. We don't have to track down pajama bottoms or uniform shirts because they all go in HER hamper. She seems to really LIKE knowing where everything goes and having very simple systems/habits for things. VERY simple but they make our lives so much easier.
If I think she's going to be tempted to lie about something, I don't ask "did you forget to blah blah blah" - I tell her what I know (I see you didn't put away your toys last night) and tell her what to do about it (you need to put them all in the yellow basket in your room now). It seems to eliminate a whole lot of lying to save face or avoid punishment. At the end of it, what I want is for her to put her stuff away. "Catching her in a lie" on top of it isn't going to do anything except create another situation I have to punish for.
Does he do well with simple routines and habits? I find that eliminates a lot of chaos and fighting.
Does he get plenty of time outside to be wild and get rid of pent up energy? What is his diet like?
I would really try to figure out what the core of the issue is. It sounds like he is easily over loaded with input and chaos. Soccer may not be his sport. Something individual like running or swimming might be better. He may really be dealing with things he can't help and forcing him into provoking situations won't help him get better about dealing, they'll just continue to over load his already maxed out system.
If he does well with video games - maybe it's because they require total attention and problem solving. Does he also like things like Lego and building/strategy toys and games? He may need that quiet concentration to sort of calm down from the day.
I hope you find something that helps. Talk to his pediatrician and if you think he needs an eval, put it in writing. There may be things they can do for him in the classroom so that he's not SO over stimulated and maxed out on input but the system is not fast. A good teacher will start to work with you BEFORE you have an IEP or 504 (different districts seem to weight one or the other more heavily) in place.
My about to turn 5 yr. old has some difficulties as well. I do identify with your "he has to learn the hard way" statement.
Have you looked at books about sensory integration issues in children? Maybe an evaluation by an Occupational Therapist may help. Also, consider a Montessori-based environment and/or homeschooling to provide the structure he needs.
We're also eliminating dairy. Because we're combining that with other behavioral modifications, I can't say that it's helped, but I do notice that his asthma has been triggered much less.