I just wear my running shoes and strap in tight. I don't have time/room in my bag to carry my road shoes, pedals, and a wrench around all day and they only have two SPD pedals.... on two different bikes.... dumb.
I normally don't bring a water bottle and just drink a lot of water on days I spin. If I'm pushing too hard, I'll get off and take a break just for long enough to get to the water fountain and back.
The towel is probably the most important thing because eventually your shirt is going to be too wet to wipe your face onto your shoulder.

In your gyms do the spinning bikes have really relaxed angles? I'm almost 6'2", but I find myself sliding the saddle almost all the way forward just to get a feeling somewhat like my road bike.