Don't worry about it - I'm a first time owner too, with two of my own, and when I started I didn't have a clue! Luckily the guys on here helped me out
The reason why fridging isn't recommended for wound cases is because fridging slows down their metabolism - good if they've swallowed something potentially dangerous, like pebbles, not so much if you need them to get nutrients from their food.
Any tank will do fine - some owners even use a large container! - as long as he can stretch to his full length and is kept somewhere safe where people won't knock it over. Be aware that if this tank isn't cycled (which most tanks that aren't in constant use are) you will need to replace the water with dechlorinated water daily - this is why some owners use large plastic tubs, where the axolotl can be put in another tub, the old water replaced and the tank ready for use again with ease. I use a small 'nursing tank' which used to be used for raising goldfish fry, which I replaced daily while Toothless was poorly.
Which dechlorinator are you using? Stress Coat + is very good for wounds as not only does it clear the water of chlorine and chloramines, but it also has aloe vera in it, which keeps wounds clean. I use it for all my tank water
Now, 'dashed' does mean 'made up with hot water,' then it's added to the water when it's cool. Your axolotl should be fine in it 24/7 - remember, they don't 'breathe' the water, they take in air when they come to the surface and store it in their lungs. After a week, do a 60% water change (if you are replacing water daily because of an uncycled tank you will need to make up smaller portions of the tea bath daily depending on the size of the tank, and you won't need to do the 60% water change because you'll be doing 100% water changes every day). You basically replace the tea dosage when it runs out - if you have a large, cycled tank you can do 60% after a week, and then smaller changes every week after - if your baby still has wounds after this time you can start the process over again. If you need to replace water daily, you will need to replace the tea bath daily. It all depends whether the tank you are using is cycled or not.
I don't think it's contagious at all. I would only be concerned if I saw cottony tufts sticking to the wound, in which case it may be fungus - that is contagious as far as I know. Wounds aren't contagious and I think the mild purple discolouring may be from the medicines rather than a signifcant infection. I'd use the tea baths until the wound disappears and watch for fungus/unusual discolouration, spots, etc before putting him back in his tank. It looks to me like he got a small wound and was a little stressed out by the moving, temp changes and treatments - don't worry, I stressed mine out big time when I had to cycle my tanks in much the same way. Be sure you float him in to the new tank slowly if he's been in the fridge as he might be temperature shocked otherwise - simply remove him from the fridge, let him get used to room temp, then float him in to the new 'nursing' tank.
I'm not sure which plants you are using so I can't say whether they are safe. You could take a picture and put it on here to be ID'd or look for it online - I found my floating plants this way (Mini Water Lettuce, axolotl safe, but invasive). If you want, you could also post a picture of your baby so we can see how he's getting on
So, action plan - get nursing tank ready, prepare tea bath, get him up to room temp, slowly introduce him to the new tank, add tea bath, then prepare dechlorinated water if the tank needs water changes daily.
If any of this doesn't make sense, please feel free to ask. I didn't get any of it when I first got my Tooth either