OP-its obvious youre not a noob, but i just know that ick looks like you rolled the fish in sugar. if your fish dont look like that, i dont think its ick. you mention white feces, that usually means parasites. you mentioned you recently restocked the tank? if i understand that right, it would have triggered a mini cycle, i.e. causing the ammonia to jump and nitrites.. the bacterial load was too high and its compensating by creating a cycle. also, restocking or moving fish causes stress=vulnerability to disease.
I do NOT believe this is ick, and i wouldn't raise the temp (just fyi-to treat ick just raise the temp and use aquarium salt-dont ever waste money on ick meds) and add metronidazole (commonly called Flagyl). it is used to kill parasites, and is pretty common and cheap. add it in the food is the best way-as it works better if the fish ingests it than just floating in the water column. it will make your water blue green. ive found that metro works well.. but you do have to use it for the full time.
as a conditioner, you may want to start using Prime-its really the best out there that ive found. and since you do have ammonia-and any slight amount of ammonia is quite damaging to fish i would do a 50 percent waterchange before dosing with anti parasitic meds.
the gill movement and rapid breathing is due to stress and trying to compensate for the ammonia damage-and just being sick in general. the airstones dont really add O2 to the water per se, they just break up the surface area.
everyone on here has different advice as to what works for them-ive dealt with parasites and infections, so this is just what has worked for me. keep in mind that most meds used will kill the beneficial bacteria in the tank, so just be prepared if your parameters change..
right now, your number one priority should be addressing the ammonia problem-as it will kill your fish. and you do that thru waterchanges, then you worry about the disease.