Hi, Mindy, welcome to FT!
Have you looked through the Japan forums here, too?
I haven't been specifically to Kyoto, so can't help you there, but I will say that I've always felt safe as a single woman walking at night in Tokyo. Just keep your wits about you, just as you would anywhere else.
If there are rules against running at the shrines, they will likely have some sort of signage as Kyoto is a tourist destination and they are used to foreigners. You can also ask at your hotel.
As far as fashion goes, will Japanese clothing fit you? Even with "free size" clothing items, I didn't always find items that were long enough for me (5'8", size 8 back then). Shoes weren't so much of a problem, although they weren't always wide enough.
Enjoy your trip