Powdered detergent is typically preferred because it rinses out better. I've read that the detergents (sodium lauryl sulfate, etc) that typically make up liquid detergents can clog the pores of membranes, but I'm not sure if that's true? Atsko sport wash could be good, but it's probably just liquid detergent and nothing else, so if detergent cloging pores is true then maybe it's bad? I can't find any information about what it's made of except that it says biodegradable vegetable surfactants on the bottle, and I think that just means some variation on sodium lauryl sulfate. I've used Atsko sport wash extensively, but I don't have enough ptfe membrane clothing to know how it affects it. It works fine for everything else, and it's good at getting out smells, but it's not as powerful as some detergents.
I personally try to avoid anything that doesn't tell you the ingredients. And definitely try to avoid anything that might have stuff that is meant to be left behind like softeners, fragrances, optical brighteners (if it looks blue it has brighteners), etc.
Event recommends frequent cleaning since the membrane is not well protected from dirt and oils. Even a little laundry detergent residue is probably better than it being really dirty.
The dryer should be fine, but high heat could melt the seam tape off. Read the care label to be sure.