If you decide to apply for citizenship, they might look at your work history and see you were with your first employer for 1-2 months They might think you "used" that employer, and took advantage of them to get a green card. It wouldn't be fraud necessarily, but it would be suspicious at a minimum (in the eyes of USCIS).
If, however, you had documentation showing that the employer didn't hold up their end of the deal (less pay than you were offered, making you work w/out pay, all the things you have mentioned), then USCIS would have no reason be to suspicious.
I'll put it like this: USCIS can assume the worst, and it is your job to make it clear that is NOT the case, and you did everything you were supposed to do.