While you are trying to cycle a tank you should not use antibiotics, the most common treatment for fin rot which has gotten beyond the very mild stage or which isn't responding to water changes of mild treatments like MelaFix. The nitrifying bacteria can be at risk, especially before the biofilm in which they live and which can protect them from some antibiotics has not had time to build up.
I would be less concerned with low levels of ammonia or nitrite while cycling with fish, especially if you have a tank pH in the 7s. It is important to hold the ammonia levels at least to .25 ppm as shown on the API ammonia test kit. This should help to keep the nitrite from getting out of hand while still encouraging the cycle to progress at a decent pace. However, because of the nature of cycling and the bacteria involved, ammonia can fall to 0 well before nitrite has peaked. So you should monitor the nitrate levels and if they start to move up and/or the fish are showing distress, water changes would be advisable to control nitrite levels. It is very rare that .25 ppm of total ammonia in a tank has sufficient NH3, the toxic form of ammonia, to be a concern over the short term (a week or two). Most of the ammonia will be in the much less harmful NH4+ form.
One walks a fine line when fish are in a tank during cycling. On one hand there is the desire to minimize any potential harm to the fish by trying to keep the ammonia and nitrite levels down and the fact that the lower these levels are held at, the longer it will take to complete the cycle. So the choice is a bit higher levels for less time or lower levels for a more extended time. Either one can potentially be harmful to fish. However there is a limit to how high either level can be before they will certainly harm or kill fish. For me .25 ppm of total ammonia is OK in most (but not all) cases of cycling.
As for fish in an uncycled tank, you can do this indefinitely as long as you are changing a lot of water. In your case with a goldie in a 10 it could be daily. I would suggest you feed as little as possible in all your tanks until they have cycled and you can return the goldie to her normal home. Because you have it in an uncycled tank, you could treat the fin rot with antibiotics as long as you do the water changes before you dose that days medication. In addition, the daily dose should be the normal one plus something additional to compensate for the water you changed.
That is if you changed 25% of the water, dose one and a quarter times the daily dose to compensate- if you change half the water dose one and half times the normal daily dose. In essence you are replacing the meds removed as well as dadding that day's new dose.
If you feel better about trying to keep both nitrate and ammonia at undetectable levels in a tank that is cycling, be prepared for it to take much longer than normal to complete the cycle. However, this is the worst that can happen. Making a mistake in the other direction and letting the levels get too high can be a much more serious problem.