I've run my trainer with nutted axles and it wasn't a problem. At least on mine, the cups ended up clamping onto the frame with the nutted axle keeping the bike from dropping down.
As for QR in front and non-QR. That was more common years ago when horizontal drop outs were the norm. Allowed the rear to be tightened so it wouldn't slip and the front wheel (the one more likely to be removed when transporting) to be taken off without tools. Not so common any more. on the positive side, if it is a freewheel-equipped bike, you're less likely to snap a rear axle.