Just to clarify--are you planning to enter only "mare and foal" classes? Or are you meaning that you are going to bring the mare & foal to the show & show the foal in the foal classes?
I ask because not all AMHR shows offer "mare & foal class"--as I recall that is an optional class, and falls into the combined division A & B classes, for mares 38" and under. I'd probably be safe in saying most shows do not offer it--I know our shows here do not include that one. In that class the mare & foal go into the ring together.
The regular foal classes are just for the foal--the mare is not allowed into the ring when you are showing her baby. So--you will have to leave the mare in her stall or have someone hold her outside the ring (which can often mean she fusses and hollers and manages to get the foal upset--foals often show better if mom is left back in the stall so baby cannot hear or see her). The foal should be halter broke and leading well enough that you can lead him places on his own. He should be able to set up a little bit--if there are others in the class there's a good chance that some foals are going to behave and show better than others, and the judge might favor the foal that actually stands up and gives the judge a chance to look at his conformation. Overall though I think judges are probably more forgiving of foals--but it's still nice if baby will lead along quite well & show himself off a little when stood up.
Obviously the mare's condition won't matter if you are showing only the foal; for mare and foal classes, yes, since the mare is being judged her condition may count toward her placing. According to the rule book the mare and foal class is judged 100% on conformation and uniformity. Obviously a broodmare isn't likely to be as svelte and well conditioned as a mare showing in the regular classes, but at the same time a very potty "well sprung" mare probably isn't going to score as high as a more streamlined mare of comparable quality.
A broodmare with foal at side can be entered in either broodmare or the regular aged mare class--but either way the foal can not go into the ring with her.