Yes, you got it right. He would start his enlistment right after his tour...
It's a shame that he can't be on active duty...instead, he will basically do office work... About his "surgeries," there wasn't enough transparent information given to the public about these...in fact, there had been some impressions of secrecy surrounding his "injuries..."
He did a livestream on weverse yesterday, in which he showed the 7 tattoo on the back of his left shoulder, and right below it a scar from the surgery is visible. His shoulder injury is from a predebut incident in which he was involved in a hit-and-run. A car slammed into him, then ran him over and his shoulder was dislocated, plus there was muscle damage. I think the secrecy you're referring to is how he hid this accident from the company, out of fear of being dropped. The company has claimed they never would've dropped him, but I think they would have.
The only thing that I think is super weird, is when YG talked about this in a video of them all eating together. It's weird to me cause a camera is there. His bandmates, other people who know him well, describe him as private. Look how long it took him to show off the 7 tattoo, just as an example. It's not even on a revealing part of his body, it's his shoulder, and he's still modest about it. I think the company, who obviously planned to film their meal together, told him to talk about it. And like, I think it really did happened, there's tons of evidence, but it bugs me that they encouraged him to talk about it just to get people more invested in him. YG strikes as the type to fall for that, cause there have been many instances of "PD said I should, so I did". Like for whatever reason if it's just logical enough YG will fall for it and do it; such as being told people will get more attached to him if he shares something personal (and that's not wrong, personal stories absolutely do get fans attached. That's how shows like AGT manipulate viewers into liking an act and keeps people watching).