Welcome, poor guy is in need of a little more TLC :) But he'll be getting it now since you are willing and learning :)
Anything 1 gallon and larger is idea for them- just keep in mind the larger the easier it is to heat.
Depending on the size of the tank you will be getting, the heater can vary. I would just recommend finding a 20-25 watt adjustable one, as those are easier to keep the temperature over the preset ones. Make sure to pick up a thermometer as well to keep an eye on the temp.
You will need a water conditioner- something to remove ammonia, nitrate/ites and binds metals. Most conditioners will do all that- as mentioned above, Stress Coat is a good one. Seachem Prime is better.
Spring/bottled/distilled water is not idea for fish- as it lacks the proper nutrients and minerals needed that the filtration process removes. It's safer and healthier to use tap water with conditioner.
Stop Melafix- any of the "fixes" can easily harm the labyrinth organ. I disagree with the statement above that you should continue to use it- as in this case it's not something you need or should be using. Not to be rude, but the treatment mentioned above will do more harm then any good. Only reason I am suggesting something other then clean water is because he is pretty far with the rot and melt.
He seems to have fin rot and fin melt- both easily fixable with two things: clean water and AQ salt.
What you will be wanting to do is get a gallon water jug, fill it with warm tap water, mix in 2 teaspoons of Aquarium Salt (per gallon), and water conditioner, making sure the salt is completely dissolved prior to putting your guy in it.
Remove the guy from the vase using his cup and empty the water from the vase.
Pour in the treated water (make sure using a thermometer it isn't too cold or too hot.. do this while filling it up- 78-82*F is what you are aiming for).
Float the guy in the water in the vase for 20-30 minutes, all the while every 10 minutes or so removing some water from the cup (dump it down the drain), and add in some of the treated water into the cup to acclimate him to both the temp and the chemistry properly.
You will need to do daily 100% water changes for no longer then 10 days.
During that time, the salt should kill off the rot. After the 10 days you will just want to be doing frequent water changes to try to reverse the melt.
Depending on what you get-
1-4 gallons filtered
you will want to be doing 1 50% water only and 1 50% substrate cleaning per week. For unfiltered
you will want to be doing 1 50% and 1 100% water change per week- always dosing with the conditioner.
For 5+ gallons filtered
you will just be doing roughly 50% per week with regular substrate cleaning. Unfiltered
would be 50% per week, 1 100% per month- dosing with conditioner.
But for a few weeks I would add in one extra 50-100% per week until his fins heal up.
The warm, clean, conditioned water should return him to health. Just make sure that you always adjust him to the new water appropriately, and keep up with the treatment with the AQ salt for the rot.
Whatever size tank you get, keep in mind these guys love a good amount of plants, decorations and especially a cave of sorts. Having something so open and bare could be stressful.
He also seems not to be eating- he is very skinny. It is most likely due to the water temp and chemistry. So hopefully with the clean, treated water he will start eating once more. Remove any uneaten food within minutes of feeding to reduce it from spiking up the ammonia between cleanings. 2-3 pellets of the size you are feeding per meal, twice a day is recommended (4-6 pellets total a day- their stomachs are the sizes of their eyes so don't want to do too many). By feeding once a day you will either over feed, or under feed- both are potential health risks. Smaller, multiple meals a day are what you are aiming for.
Good luck and any questions, feel free to ask!