You would not ever be expected to know if you should or shouldn't apply for benefits. It is never wrong to apply, even if you are flagrantly ineligible. You do not have to read and interpret or argue or decide about any of the laws regarding this or that aspect of unemployment.
Anyone is able to apply for benefits without penalty at any time. It is the agency's job to determine whether you are eligible for them or not. Apparently there WAS a decision to approve benefits in some size shape or form made, or you would never have begun receiving checks. They very rarely make a huge mistake, and just start sending someone money without a decision, and unemployment benefits aren't just lying there waiting for for your decision that you'd like to have them.
The problem comes if you, in any way form or fashion told the system something that was not quite 100% TRUE in order to receive these benefits you have received. Okay, when you filed the claim, did you make it perfectly clear that you had last worked at the school system? Yes, there would have been no base period wages showing from the school system on the monetary determination, but you are expected to list the last date you worked for a covered employer (and yes, in spite of their rather special status in unemployment circles, schools are covered employers) If you did that, if you were perfectly honest about the situation where you quit the former job and went to the school system and then were let go due to the summer holiday, then you are doing all you can do. It is up to the Michigan system to make the eligibility determination based upon the information they have, but if you are not being 100% honest with them, then you are at fault.
There is also the issue of the weekly or bi weekly (I believe that's what it is in Michigan) certifications that bring a check. BE VERY SURE you are perfectly honest about that, about the way you answer all the questions on the certification every week. Okay, you are job attached. In other words, it is assumed that your lack of work right now is only temporary, and you do expect recall from the school system. Did you answer any questions about this in your claim filing, and did they inform you that you are required to be making job searches? If they do, ARE you making such job searches?If your claim was selected for audit, and you were called in, would you be able to tell them the names of, and show proof of places where you had submitted any resumes or applications?
In answer to your question is it possible I will have to repay the benefits, the answer is yes, it is possible. If for some reason they determine you were not eligible for benefits, you would have an overpayment decision, and they'd want the benefits repaid. If you were approved due to your providing false information to the system and got money you were not entitled to because of your falsehoods, it's a fraud overpayment. And fraud overpayments come with penalties, and will be pursued more vigorously, and could even eventually lead to criminal prosecution. Simple overpayments where people were first determined eligible, and then the decision changes are non-fraud overpayments, and though they do want the money back it is not considered a really serious matter.
The "six weeks to make a decision" is a suggestion, it's not set in stone. You are not totally home free just because you haven't heard anything different and six weeks have passed. You have probably received four or five weeks of benefits. As long as you are doing it honestly to the best of your abilities, you're fine. You are fine based upon doing what they have told you to do. As I said, BE SURE you are NOT telling them anything incorrect or based on false information. If you are, stop filing those certifications at once and contact the system again and discuss the circumstances with a person.
And do not even for one second consider filing for a week when you have gone back to work. Each week you make a certification for it is clearly spelled out what they are asking you and what they want to know. Filling the certification out wrong is fraud. If you file for a week that was Sunday through Saturday, and you file for that previous week, even though you're going back to work and your official job begins with school Monday morning, that's perfectly okay.
If you went back in to your job for even a few minutes of paid orientation on that previous Friday afternoon, when you do that certification for that week, you must report it. They don't care so exactly how much you are paid (it's possible you will not be PAID for that orientation day until several weeks later as a pretty good estimate of it. But they want to know about it THE WEEK IT WAS ACTUALLY WORKED, on your certification. You may not know exactly how much you will be paid for that amount of work but what you should do is ball park it. They're talking about gross wages. So if you went in and worked six hours at $20 an hour, you need to report that you did work and that you made $120. Otherwise, you're committing fraud. When you look back, and you have worked at your school job and made more than your unemployment weekly benefit amount, just stop filing the certifications. The claim will stop.
Do not be surprised if, sometime on down the line later, you receive a notice from the unemployment system inquiring about your situation. This is probably something that the former employer who has been tapped for your unemployment insurance isn't going to like at all. You quit them to take the school job, and now you're costing them money by receiving unemployment benefits paid from their tax accounts. At some point, they will be notified about this, and they may use their right to appeal. The system may have more questions, may look at your claim again later. As long as you have been perfectly honest about your situation, that should be fine.
That you have school wages overlapping with your wages from the separating employer and unemployment wages all during a quarter or two will probably lead to the system looking at your claim a little more diligently than something that was more cut and dried. As I said, tell the truth to the system, and you will have done nothing wrong.