I wash my riding clothes when I have the chance and hang them out to dry if conditions allow. I hang them on my panniers or under a pannier flap to dry while riding if necessary. Washing might be in a sink, in a ziploc, or even in a coin operated washing machine once in a while.

As far as frequency of washing of riding clothing, that varies. I usually have two changes of riding clothes (some times only one) so I can usually wear one while another is drying, but I also don't mind putting them on damp.

Sometimes riding clothes get washed every day and some times they go several days between washings. Washing of off bike clothes is usually less frequent and I might only have one change of those. Running shorts (with built in brief) and a tech shirt are pretty tolerant of infrequent washing. Ditto for my nylon zip off leg pants (also with built in brief).

Pretty much the same for showers. I shower when I can, but don't stress if I have to go a few days without.

I usually try to use a coin operated washing machine for the whole lot at least once every few weeks, but some times more or less frequently.