dont get too excited i'm no expert either, but i'll add a few thoughts at least! your
nitrate needs to be no higher than 20, but preferable much lower, so i guess 50% water changes every day will get them down. high nitrates (or
ammonia or
nitrite) can cause stress and thereby cause disease. also, a non medicinal way of getting rid of ich which onw of the mods on here uses is high temp (about 80 i think) and 50% water changes. the temp speeds up the life cycle of the ich, and the water changes physically remove the ich paracites from the tank so they cant re-attach to the fish and multiply. i however had dying fish (not through the ich though! i had another disease in the tank too!) so was lacking patience for this and medicated which worked in about a week for the ich. i used medifix (antibacterial - for the other disease i had in the tank) straight after the ich treatment as didnt have time to lose with waiting a week or so before introducing a new med and they didnt harm the fish mixing those 2 meds together. i think tetras can use medifix, but just a reduced dose if ever you need it - always check first!!
anyway, assuming your tetras have ich, is it tiny white dots like a pin head? thats what was on my fish and it was just on their tails. if the white dots are really small its probably ich. if there's any bigger cotton wool looking white stuff going on with your fish it's probably fungal which will need an antifungal treatment.
i struggle with whats bacterial and whats parasite and whats fungal - i guess time and experience will teach us those! i've found very little in the way of pictures of diseases. just descriptions which are all quite vague and similar sounding!
also, i use water the same-ish temp as the tank when i add water to the tank so as not to chill and stress the little fish. i use a mixture of water from the hot and cold tap and add a dechlorinator to the bucket of water too before chucking it in the tank. i cant see whats wrong with hot tap water. its all come out of the same reservoir!! the cold mains pipes are prob made of copper same as the hot water pipes. anyway, my fish seem happy with it (now i've sorted out the ich and other disease which i think was columnaris, but i will never be sure!)
anyway, ramble over! keep us posted with how the tetras are doing. good luck. who said fish keeping was easy?! i was surprised how tricky it got when they got ill!!

anyway, sorry i'm no big use. if i could help you i would!