He's 5, so canceling something that's 2 weeks away has no relevance, in a child's mind, to what happened this morning at school. He needs more immediate consequences whenever he commits an infraction - and that is classroom management.
I understand why you made him write a letter to those involved, but even at 5, kids aren't really "sorry" if you know what I mean. I agree that it made sense to put him through the exercise, and certainly to spend his saved money on a replacement toy (excellent idea), but that's not to say that he's truly sorry. That's a developmental stage he hasn't reached yet. So he can learn that there are consequences that affect HIM, but it's not enough to make him really feel true remorse. I do think you can reinforce this when he wants something new or wants more money for something - your attitude must be, "It would be nice to buy some candy, but unfortunately you decided to steal the Lego toy and so you, of course, had to buy a new one. Maybe next time you won't make that decision." Keep the emphasis on each decision that he makes, and tie the immediate consequence to that.
I have no idea what the principal means by "last chance"!!! They are not going to expel a 5 year old, for heaven's sake. So find out what that means. So sitting on the rug, he needs to keep his hands to himself - but that's a common lesson every day in preschool, kindergarten and first grade. A lot of schools use carpet squares from the remnant store for that very reason - to keep kids in their own space. It would seem to be a very simple solution to have him sit right next to the teacher and maybe have something on the other side - a chair, a stack of books, or just a bigger space. If there's a teacher's aide in the class, that person can help. If there isn't, ask if there is a floating aide who can be in your son's class at a specified time when a problem activity might be involved.
I think it's perfectly fine for him to have to play at recess near the playground monitor - but he does need to get out and run off his energy, so I don't think he should have to stay in.
I do think a system of rewards is good. You don't want the teacher to have to do this with all of 20-25 kids all day, but many schools have a sticker chart or a stoplight/clothespin system (clothespins on the green light portion of a poster, first offense the clothespin gets moved up to the yellow light, and 2nd offense it goes to the red light when privileges are suspended). In my school, the kids moved their own pin up to the next light, so it really imprinted on their brains. You could have a small reward at the end of a certain number of stickers or good reports. Work this out with the teacher - I'm 100% positive this has happened before with other children in other years!
I think canceling his birthday is not a good idea - I think it sends the message that he's not lovable or deserving of a birthday. Don't think about the embarrassment in front of other parents, but do think about the free-for-all environment of laser tag and whether that's a good setting. Also, have other adults on hand so that you or your wife can immediately take him out in the event of a problem behavior, and still have enough supervision of the other kids. Of course, he will have a meltdown if you take him away from that, so perhaps a lower key party is good. Have you ever done one with the old fashioned games (sack races, potato/spoon relays, etc.)? We used to do those for our kids, and I got older books out of the library with great, simple and inexpensive ideas. Kids LOVED them because they were such a nice break from bounce houses and ball pit parties. Kids this age don't need big venues. (We also kept parties very manageable in terms of size - when our kid turned 6, he invited 6 kids. Period. If you are having a huge group, it's overwhelming for most kids so you need to be very structured.) However, your invitations are out, so you can't rescind them. If you change the venue and the other kids are disappointed and are only coming because of the activity and not your son, then they aren't great friends. Or they can learn to adapt. At home, it's easier to remove a poorly behaving child, even the guest of honor, for a quiet time in his room and let the other kids continue with games. If he wants to return to the group (which he absolutely will), then he'll settle down. Take him out every time he's physical with other kids. Every time. Again, have other relatives or friends there to run the party so you can be in charge of your child. If, however, you are committed to the laser tag party, then have a system to remove him from the party and take him out to the car (zero stimulation) every time he is inappropriate. It's energy-intensive for you, but it will give him the message that it's his behavior that is the problem, not his worth as a birthday child.
This is so normal for a 5 year old. If you are consistent with consequences every single time, no matter how inconvenient it is for you (and it will be!), you will get much farther with him than if you talk to him about it over and over. Use a very, very brief term (like, "hands to yourself") that covers everything from pinching to hitting to even excessive hugging or high-fives). They don't hear us after a while - they just don't. And often the talking/reprimanding gives them attention, even negative attention, that they thrive on. But they absolutely get the idea that it's terrible for their OWN fun and enjoyment if they engage in this behavior and are separated from the family or the group.