fair warning, i'm going to say some things that the folks here aren't going to agree with.
first,you need to do two things with fish in cycling, do ONE 50% water change daily and keep your water temp between 80 and 85 degrees F. do not do multiple water changes in a single day, that will remove the ammonia than you need to start the cycle. yes, it's a double edged sword. you NEED ammonia to start the cycle, and you need to keep the fish healthy, it's VERY difficult to do.
***I see your point about the ammonia being needed, but the permenant harm to the fish has to be taken into concideration. I would do one water change at 75 - 90% depending on the level of ammonia.
when the ammonia levels drop off then nitrite levels will spike and again, you need the nitrite to complete the cycle, but it's VERY toxic to the fish. so again you are juggling.. keep doing 50% water changes and keep the temps up.
***Same advice as stated above.
fish in cycling actually goes faster than fishless cycling.. but it's very detrimental to the fish.
***It may seem like the cycle goes faster for a fish in cycle, but when you add more fish, you end up with ammonia problems. Fishless cycling allows you to fully stock once the cycle is complete.
the white spots you saw initially is probably ich. the shortened fins you saw are probably from ammonia burns, if your fish survive the cycling process, they will grow back. the fish weren't defective when you bought them, except that the store obviously has ich in their tanks that came home with the fish you bought. the best cure for ich is high temps 85 - 87 degrees with an added air stone and air pump (i like the little bubble walls for this, they look good, and are cheap to operate) because warmer water doesn't hold as much oxygen for the fish to breath. some folks like to add aquarium salt to help kill the ich, but the higher temps will do it without the salt. salt is a good cure for many diseases though, i think the dose is 1 tsp for 5 gallons of water. it'll help that fin heal too.
***I agree with upping the temp to help with ich, but raising the temp just speeds up the life cycle for the parasite, it does not kill it. The salt kills the ich. To kill ich with temps, you must add salt.
the big thing while you're doing these water changes is the dechlorinator, ALWAYS treat the water with dechlorinator before adding it to the tank. i should probably add here that there are some excellent products out there that not only dechlorinate but also help jump start the cycle, one i know that some fish folks use and recommend to others is Seachem Stability.
***I don't believe in the bacterial aids, but dechlorinator is a must. I've never used Seachem Stability, but I use Seachem Prime.
and the water temp, but one or two degrees won't harm the fish. it would harm the fish if you poured icewater into an 80+ degree tank and took it down to 70 degrees in a matter of seconds, that will kill your fish. pouring slightly cooler water in and dropping the temp to 78 for a few minutes won't bother the fish at all. your tank will cycle faster and the fish will be healthier if you keep the temp in the tank over 80 degrees.