Cycling isn't just having a filter. It is the process of natural bacteria set up colonies in the tank to deal with waste the aquatic animals create.
There are bacteria that eat ammonia and produce nitrite, then bacteria that nitrite that produce nitrate, and finally nitrates are picked up by live plants and excess removed with weekly water changes.
We always ask for water parameters when people notice issues with axolotls. Ammonia burns are common in un cycled tanks and tanks with a cycle crash. The numbers for ammonia, nitrites and nitrates can also tell you and us what part of the cycle you are at. After the tank is cycled and established it's reccomended to test the water every week before the partial water change.
Gills should look long and fluffy, but some genetics and situations can cause shorter or balder gills. But their gills are a good sign of health.
An axolotl who was purchased with fluffy gills then loses those gill feathers might be ill. And the water quality would be the first suspect. Ammonia and nitrites are toxic to axolotls, anything .25ppm for ammonia is bad but you want to aim for 0 since that means the tank is cycled. Nitrites are toxic over zero as well. Nitrates should be under 40ppm and can be removed with 20% water changes weekly or twice weekly depending on the waste load. Waste load doesn't just mean the amount of poop either, they secrete waste constantly in other sources too.
They prefer pH between 7.4 to 7.6 however they'll tolerate 6.2 to 8.0. There is a high and low pH test included in the freshwater test kit I own. I also have a medium range pH test that I use occasionally. Higher pH makes ammonia and nitrites more toxic.
They prefer harder water too and not soft water.
Tap water is fine if you have a good dechlorinator. I use one that removes heavy metals too because my local water has dissolved metals in it. Look for something that removes chlorine and chloramines to cover all your bases.
Another major thing to worry about with their water is temperature. 75°F and above can be dangerous for a few days. They are a cold water species. They need temperatures under 70°F. Higher water temps increase sickness and make them very susceptible to fungus. Younger axolotls are more sensitive to water temperature and quality.
Getting mesh lids instead of glass, using a hang over the back style filter instead of in tank filter, installing a spray bar or something to move water at the surface, and fans are all ways to cool a tank without purchasing an expensive chiller. You can also rotate frozen jugs of water but this can cause a severe swing in temperature and do more harm than good. I had a rotation schedule for my frozen bottles during a summer heat wave. Moving the tanks away from the sun and to a lower level will help too. Fans alone dropped my tank five degrees F.
Avoid 100% water changes in the tank because the disruption of the natural bacteria is high.
Adding bottled bacteria can help with the start of a cycle. Cycling with the animal in the tank will need daily water changes of about 50%, 70% during spikes, and the process will take over a month. It's a pretty intense amount of work.
Unfortunately I don't shop online much so I can't link a cheap test kit. I use an API Master Freshwater Liquid Test Kit. It works very well and is more accurate than test strips.
http://www.axolotl.org/requirements.htm
This is an affiliated site to Caudata.org and has lots of good information on the cycles and water quality preferences. It's a great resource to see if you're on track. They have other information too from food to breeding to tank suggestions and filter recommendations.
