USA to Frankfurt (FRA), change airlines to Dresden and 2 separate tickets

Three hours would be enough time to complete this at any airport I have been through--if your flights run on time.

The only concern here is that, on two separate tickets, you will be responsible for the cost of a replacement ticket if your first flight is late and you miss your second one. This cost could be considerable on the way home if you needed to buy a new transatlantic ticket at the last minute.

For this reason, I wouldn't be comfortable with a gap that short between the flights. Three hours is no problem, but it wouldn't take much of a delay for making your next flight to become a challenge. Ultimately, it is up to you how much you are willing to risk buying a new ticket.

Note: travel insurance will typically not cover such a missed flight, either.

You have three hours for both connections - that's tons of time. And since you won't be clearing immigration and customs on the return trip (you'll do that at the first US airport at which you arrive), you'll have, in effect, even more spare time.

I wouldn't worry about this at all.