In a nutshell, aquatic life creates waste (i.e. faeces, uneaten food). This waste turns into ammonia. A colony of bacteria feeds on ammonia and then convert it into nitrite. A different colony of bacteria feed on nitrite and convert it to nitrate. Nitrate is the end part of the nitrogen cycle. Ammonia and nitrite are very toxic to axolotls and other aquatic life. Nitrate is also toxic but less so and axolotls can tolerate low levels of nitrate. To keep nitrate at safe levels (below 40ppm), it is important to do regular water changes. You would usually do partial water changes (say 25% of the water each week or enough to reduce the nitrate to a safe level). However in an unicycled tank or whilst a tank is establishing a cycle, there is not enough bacteria to consume the ammonia created from the waste and so ammonia and nitrite levels (depending on what stage your tank is at in the cycling process) rise to potentially harmful levels. Cycling is the process of establishing these important colonies of good bacteria. In a cycled healthy tank, the readings should be 0ppm for both ammonia and nitrite and some reading of nitrate but kept below 40ppm.
There are two methods of establishing the cycle in a tank. You can do what is called a 'fish-in' cycle where you cycle the tank with your aquatic life (in this case, axolotls) in the tank (therefore the waste they produce is the ammonia source). The important thing to consider is that you risk potentially exposing your axolotls to harmful levels of ammonia and nitrite during the process if you are not diligent enough with water changes. You need to very closely monitor parameters and ensure that you are maintaining ammonia and nitrite levels at a safe level and do water changes as much as needed to keep the ammonia and nitrite levels at 0.25ppm.
The other method is called a 'fishless' cycling method. This involves using an alternative ammonia source and not keeping any aquatic life in the tank until the cycle is fully established. You would keep your axolotls in tubs (doing 100% water changes in the tubs daily) until the tank is fully cycled using an ammonia source such as pure ammonia or some sort of fish food. This way means you can crank up the ammonia and establish the cycle without worrying about exposing your axolotls to harmful levels of ammonia or nitrite during the process.
Now in regards to your situation specifically, I suspect that the cycle has not had a chance to establish properly given you routinely have been cleaning everything out thoroughly. When you do this, you are likely removing a lot of the good bacteria and interrupting the cycle trying to establish.
Sudden temperature changes can be stressful and acclimatising them is important. However I would doubt that one incident would have resulted in your current situation. If you are taking the axolotls out each time you are cleaning their tank, this could also be stressful for them, especially if doing this often.
Some have different methods. This is how I do regular water changes:
1. Have all your buckets and everything you need ready and at hand and test the parameters.
2. I remove the ornaments I have (just as they get in the way) and place them in buckets
3. Stir the sand up (this is good to do regularly to prevent noxious gas pockets building up in the sand)
4. I use a syphon to gently hover just above the sand to pick up debris (poo, dead plant material etc) but without sticking the syphon into the sand as this will suck up the sand. If you gently hover above without touching the sand you should be able to pick up most debris without taking out too much sand. Whilst syphoning into the buckets, I keep an eye on where my axolotls are so to avoid them as care must be taken not to accidentally syphon too close as syphons can cause injury. But as long as you are diligent, they don't usually cause issue.
5. Once I have removed enough water (I gauge how much to take out by what my test parameters were before the water change (so for example, if my nitrates were at 40ppm and I wanted to reduce it to 20ppm. I would need to remove 50% or half of the water), I very gently rinse out my filter media in the buckets of aquarium water (it is important to ensure that you don't rinse items in chlorinated water as the chlorine and other chemicals additives to tap water will kill off the beneficial bacteria and affect your cycle). I only rinse them very gently and I usually only rinse the filter every couple of weeks (different setups and filters may need different levels of maintenance).
6. Ornaments are returned to the tank and the tank is refilled with fresh dechlorinated tap water.
Have you noticed your floating axolotl has done a poo at all since?