When it comes to Ammonia and Nitrites, the cheap API test kits will work fine, since there should be NO detectable Ammonia or Nitrite in your tank. I would recommend a high quality kit for Nitrate. Salifert and Red Sea really make good kits. In the meantime, I recommend doing several water changes to be on the safe side.
Whenever you transfer stuff from one tank to another, there is always going to be a disruption to the biosystem. Stirring the sand can release hydrogen sulfide, laying rocks in a different position can cause die-off of microfauna, or any number or combination of things can happen. It's always best to transfer everything over to the new tank first, let it cycle, and then transfer the livestock. If your fish are suffering from a parasite or fungal infection, it likely wasn't caused from the move. However, less than optimal water conditions and the stress of moving the fish can weaken their immunity systems enough to succumb to a disease they might normally be able to ward off. Scratching is a classic symptom of parasite, which is usually Ick. Puffy eye(s) is usually a sign of popeye (exophthalmia), although the eye is usually grayed over, and not fuzzy. If you find spots that look like grains of sand stuck on your fish, remove them immediately and treat for Ick. Without knowing the cause of popeye, it can be difficult to treat. Poor water conditions can cause this, or it may come from a simple injury to the eye. Close-up photos (that can be enlarged on this site) will be very helpful in identifying your fish's ailments.