I wash tea towels and towels together at 60. I do one wash of them a week and the machine is full.
I do a dark wash with dark clothes, black knickers and black bras when I have a machine full and wash them at 40. I do a white wash with light clothes, white knickers and bras, again at 40.
I do one wash a week of bedding at 60.
I would never ever wash underwear or any clothes for that matter with towels or tea towels. I know it's silly but I just think it is skanky.
My neighbour often has a line of washing which will have, say, 1 towel, 2 tea towels, a bra, some knickers, a couple of tops, some trousers, socks. A complete mixture of colours. I find it really strange and can't help wondering if it accounts for her towels looking so dingy. Most of my towels are 10 years old or older and look bright