Do their showers have dressing rooms where you can keep your clothes dry while you shower, and not have to dress in front of everyone?

Yes, they do have 3 dressing rooms.

They do allow transgenders in womens area and have had problems.

Not sure. Never showered there, just use spa membership. Please inquire within.

Dressing rooms are always a few feet away from showers at planet fitness. You can change in the shower stall but most people dont. I do. Every once in a while i hear a nasty comment. Another trick would be wearing a towel. Most people store their clothes in changing room while in shower. Wear a towel in between. Changing rooms directly opposite so you can keep an eye on them.

Yes. Its very organized and private. They also have lockers.