When my son started kindergarten he wanted to stop going into the women's room with me. So I would stand outside the men's bathroom and wait for him. If he was taking too long I would open the door a little bit and yell in. I told him to use the bathroom and wash his hands, if anyone tried to touch him at all to scream for me and I would come right in.
As for showers, my 4 year old washes herself and my son was nearly 6 when he started washing himself. I would just show him how to wash and then have him do it with me washing and once I knew he was doing a good job I let him do it himself. For both kids (4 & 7), I turn the shower on for them, get their jammies and towels out for them and let them wash. With my oldest I have to check to make sure he's washed and then have him shut it off and he gets himself dry and dressed. He would take an hour long shower if I didn't nudge him to shut it off. As for my daughter (4 year old), I turn everything on for her, get her sponge down since she can't reach. I get her out and dry and help her dress.
As for chores, both kids do weekly chores. My oldest swiffers under furniture and has been doing this for 2 years. He also has to keep his room clean and the basement (toyroom) clean. My youngest cleans all the doorknobs with a clorox wipe and keeps her room clean. Both kids are expected to help set the table, clean up their own plates.