To answer your other questions, even if it gets better, you can't serve. If it has EVER appeared on any medical report, even childhood asthma disqualifies you.
And on that note, if you goes into a private doctor's report and they find something, one of a few things can happen....1) You can hide it from the military if you pass through MEPS. Then they can find it later and you can be article 15'd or court martialed for false enslitment or falsifying a government document (incidentally, the same as when they ask you if you've ever done drugs. If you say no and they ever find out....you're screwed). 2) It will just get to the military anyway, they do ask for that stuff occassionally - old medical records and such. That can happen before OR after you're in with much different consequences for the two. Either way, you end up not in the military. Bottom line is that if you know, TELL THEM. DO NOT TRY TO HIDE IT. That is by far the dumbest thing that you can do. You are risking a lot if they ever find out.
Now with that said. I AM NOT, REPEAT, AM NOT saying that you should not get it checked out. IF you feel that it is impacting your health and may be a long term problem or cause long term damage, you have the duty to yourself to find out from a credible source what it is.
I have the same problem, even with running every day. When I push myself too hard, I have the same symptoms. Now, my normal runs....no problem. But it has gotten MUCH better since I started running regularly and knocking my time down....so basically I can push myself farther before it is a problem. But I also have recognized through many repetitions that it only happens when I push too hard and is not a recurring problem. I am also willing to deal with not knowing if it is something worse. I'm not getting an MEB (Medical Evaluation Board) and be Medically DQ'd over something like that. I've known three people who got kicked out over a single doctor's WRONG opinion. Even when refuted by other doctor's, they were gone. I knew another guy who got kicked out because a record of childhood asthma surfaced - and this guy was one of the fastest runners I've ever known, with no adult symptoms.
Bear in mind that asthma is just one possibility, it could be any number of things, you could be getting a cold. See what a doctor says, there's no point in worrying about what it might be.
Even the cramping next day can be caused purely by pushing yourself too hard. When I ran my fastest run time, my rib cage was sore for about 3 days, just like you describe. I think that I have the exact same problems. But I have never had asthma nor do I ever want to go to a doctor specifically asking about it. I know my limits, I know what happens when I push them.
As Morph says, it can be anything....personal opinion, you're pushing yourself too hard.