Yes, if you see one red mite, I guarantee there are a whole bunch more you haven't seen. As for the bug you saw on your chicken, it could be red mite but could just as easily be a louse.....there are several species that prey on birds. Either way, if you are unwilling to use chemical means to be...
Red mites are not going to be eradicated by just washing I'm afraid. They wedge themselves into the tiniest nooks and crannies and will resist even concerted efforts to scrub them out!
If you have red mite, I would recommend soaking your coop in one of the many chemical solutions available as...
Have you checked for red mite? The little crawlers come out at night and feed on the birds while they sleep. This puts the birds off going in, knowing they are going to spend the night itching and being bitten!!