I dont see how testing the water will affec tthe cycling, all you will do is take a teeny bit of water out... And you need to moniter the levels of Ammonia, nitrates, nitrites, etc etc in order to find out if/when your tank has cycled. Go ahead and get a test kit. (You can print out the online page of the petsmart -or is it petco??- and take that in and save some $$)
Your danio may have a swim bladder problem. Do some big water changes, thats really all you can do for swim bladder probs, if this is what it is. This wont hurt your cycle, and you will need to do them at some point, when your levels indicate. (Most freshwater tanks require weekly water changes to be healthy)