How does heat injury’s work for cadets?

No, it does not. Make sure you let your unit know. A record of the heat exhaustion may put an extra eye on you when you're in schools, i.e. you may have instructors telling you to drink water more often. However, not notifying anyone can have adverse actions when it happens again, and they find out it is a repeat incident. You are always welcome to talk your unit's medic, or your UA for more information.

First thing you need to do is figure out if you actually HAD heat exhaustion. You say you were checked out - what was the conclusion? What actions did the (I assume) medic take? If they checked you out and said "nah, bruh, you good" and let you continue training, then no harm, no foul, drink water. If they went into heat injury remediation, you probably had a heat injury. In that case, follow the advice already given by SGT Javier Silva and CPT Brad Wilson .

I never tried to commission, so I can't speak to that part. But I was both a prior heat injury AND a prior cold injury. Never stopped me from going to any school. (I was also a weak swimmer - I got a rainbow on my gear: red, blue, green). There are a COUPLE schools (Ranger comes to mind) that may restrict specifically WHEN you can go, but not WHETHER you can go.

No but it needs to be documented and you will probably get to wear a little red tag on your gear every time you train in hit areas because they say if you have been a heat casualty once you are more likely to be one again