Question: Cycling and filter advice please.

Right now I have a 4 inch axolotl in a 5 gallon tank. I test the water using paper dipping strips and do a 50% water change every other day. The water parameters are always in the "ideal" to "safe" range according to the strips.

I plan on moving the axolotl to a 12 gallon tank that I have been cycling with goldfish for a month. At first I tried to put the axolotl in the 12 gallon but during a nitrite spike he was visibly stressed so I moved him to the 5 gallon where he is happy and thriving (and quickly outgrowing.)

So far in the 12 gallon my readings for nitrate have always been 0, althought I am getting small readings for ammonia and nitrite. I got a large ammonia spike in the first week, followed by a large nitrite spike (when I transferred the axolotl to the small tank) as expected, but after 21/2 weeks, no nitrate. I have a lot of live plants in the tank, could they be keeping me from getting nitrate readings?

Also, in both my tanks I have sponge filters. They are both starting to look pretty dirty with spots of orange (the poop color) and dark grey (unknown substance). I feel like I should clean them, but isn't that where most of the bacteria live? How should I clean or change the filters without killing the cycle?

Sounds like your 12 gallon is still cycling, yes. To test the water you really should be using a liquid test kit, such as the API master test kit. Strips are inaccurate and generally can't be trusted.

The sponge filters can just be gently squeezed in tank water (never tap water! That will kill the bacteria) to clean them off, and that won't hurt the bacteria. You never have to change the sponge unless it falls apart.