Okay, the reason this was so far back is that in times of higher unemployment activity, the states sometimes have to borrow from federal monies in order to keep their programs solvent. When times improve, they work toward getting their claims revenues back up and repaying the federal government. No, not by denying claims, but possibly by raising unemployment tax rates, and certainly by various forms of recouping money that was paid in error, even four or five years ago. They'll run and re-run their fraud and cross match programs, and in some cases, actually hire collection firms to pursue the former claimants who have an overpayment on the books from some time ago. Checking your old records and cooperating with them is the best thing you could possibly do.
As far as whether they were checking everywhere you worked personally, or whether they were checking by employer, we really cannot tell. And no, there's really no way to get ahead of potential overpayments made in the past unless you have knowledge of some serious misreporting that you did way back then. You might then mention to the fraud investigators that you want to repay this, but honestly, in this case of one misreport of one week's wages, I think I'd let it go with having paid what you have paid. If they were going to do a disqualification on future claims, that would be clearly spelled out in the overpayment statement you received. It sounds as if they have charged you a penalty, and that may be it, once this overpayment is discharged, or your communications from them will inform you of further disqualifications for benefits if they are going to occur.
I'm not at all sure how effectively you can appeal the false statement week(s). Even if they'd informed you the week after it occurred, what could you say except that you put down the wrong amount of gross wages? They might possibly have let it go without penalties if you'd attempted to self correct while you were still drawing the claim, like the next week when you received your paycheck and saw what you'd actually made as opposed to what you had reported.
But in most cases, a non-fraud overpayment is pretty much claims money you drew in full knowledge you were approved to draw benefits, when the decision to approve benefits is later overturned by appeal/decision. It's not an "innocent mistake" you made in failing to report wages made during the week, or failure to report work in the correct week. They pretty much expect you to do it correctly at the time, or you've made a false statement, and if so, you've even on a very small scale, made a false statement and committed unemployment fraud.
That said, there's no reason you need to worry about it further. The overpayment doesn't appear on your credit record or your criminal record or your "permanent record" or anywhere else, you've paid it off, and it's been dealt with.

