I like to hand-wash when possible, but sometimes I'm forced to go to a do-it-yourself carwash. I don't use any of the brushes, just the power soap/water and power rinse. Can this cause any harm to the finish or wheels?

Just bring your own bucket, soap, wash mitts and chamois and you will be able to hand wash your car there. I do it all the time since moving to Houston for school and moving into an apartment even though the signs state no bucket washing. At least I'm spending my money there so what does it matter? I also try to go at times when it isn't so busy since I take a while with my car and don't want to hold anybody up waiting for a bay.

As for do-it-yourself carwashes being hard on the paint, as long as you hold the wand a good distance from the vehicle's finish it'll be safe. To avoid grinding in loose dirt on the paint with high pressure water give the car an initial pre-rinse on the low pressure soap setting to remove as much dirt as possible before you apply the high pressure water. Put some soap and water in your bucket and hand wash away.