Will my lotls be okay in plastic containers?

So I checked my nitrites this morning and they had spiked to 5.0ppm! Ammonia is 0 and I can't recall what the Nitrates were but they are present. I've had issues with my tank cycling from the beginning (my own fault due to getting axolotls before I was entirely sure it was cycled) so I've been doing daily water changes which has kept nitrites down to .25 up until now.

I filled a five gallon bucket with ice cold water and dechloinated it and then put each of them in a small plastic container of the water. I emptied a kitchen cabinet and put them in there to keep my cats away from them. Will they be okay there or will the water get too warm at room temp and make them sick? I also have a container of the dechlorinated water in the fridge to do water changes but I'm not sure if going from room temp to very Cold water would be very good for them either? Should I cut all this out and just be keeping them in the fridge? They are juveniles and one is seems like he may have a spot of fungus / has a low appetite so I wasn't sure if I should fridge them and decrease his appetite even further. Also what is the best way to do a 100% water change? Should I use the net to move them to different containers? I feel very guilty and stupid for not having my tank cycle correctly in the first place and I don't want to put my axies thru any more stress. any advice appreciated.

Ice cols water will put them in thermal shock if they've been at room temp.
If you have spare container moving your axie is the easiest thing to do.

But in all honesty you can leave your axies in the tank and do a 90% water change to bring the nitrites down. Treat is as a fish-in cycle with daily water changes, it will be less stressful for your axies that keep moving them. If you have nitrates then you're nearly there.

You've got good advice already. For future reference, if needed. - When necessary (such as cycle crash or tank leak disaster) I put my axis in food safe bins, and load into a cooler with ice packs. they are quite happy there until tank issues are resolved.

So I tried both methods here and neither seems to have made a ton of difference. I tried taking them out for a few days and letting the tank cycle (no water changes, I installed a bubbler and added some volcanic rock). This seemed to help as the nitrites when down to .25 and my axies seemed pretty miserable in the containers so I decided to put them back. Tank kept showing .25 for a few days before shooting back up to 5.0. I've been doing majority water changes every day (taking out everything except for enough to cover them) but every morning it tests at 5.0 (could also be 2.0, hard to tell but still bad!). I can't figure out what I'm doing wrong. Most days ammonia is present at .25. Nitrates have varied between suddenly not showing present at all, to quite high the next day. Is it possible I have a messed up testing kit? I can't figure out what I am doing wrong here. I'm also going to be out of town for Thanksgiving and I'm quite worried about leaving them alone. We have an automatic feeder loaded with pellets for while we're gone but I don't like the idea of leaving them in such bad water. My axies both seem fine thru all of this, no curved tail or forward facing gills, appetite seems good as well. So confused here.

Yes be careful when moving your Axolotl from tank water into ice cold water, they need to acclimatize.

I'm not sure if you understand the cycling process fully, the appearance of NitrAtes shows that bacteria is present, breaking down the Ammonia and Nitrites. So it looks like your cycle is on the right path.

During cycling Ammonia is produced by your Axolotls waste and decomposing food, this is then broken down by a nitrogen fixing bacteria into Nitrites, and then broken down into NitrAtes. Ammonia and Nitrites are lethal so your tank needs to establish colony's of this nitrogen fixing bacteria which will breakdown your Ammonia and Nitrites into less harmful NitrAtes. The beneficial bacteria acts as a 'biofilter' keeping your water safe, all you have to do is weekly water changes of 20-30% to keep the NitrAtes down. So a cycled tank should show readings of 0 Ammonia, 0 Nitrites, and around 20-40PPM of NitrAtes.

Use these links to guide you on the process:
Caudata Culture Articles - Cycling
Caudata Culture Articles - Water Quality