Quick general info questions

Is there anywhere around Tokyo that I could make it to and back in a day that has a nice onsen with a scenic view?
How much money should I plan to have on me on average per day? I was going to use cash to hopefully avoid fee's on my cards but don't want to be walking around with tons of cash.
I was told most sightseeing places will have a sign posted saying whether photography is not allowed, I will still practice proper ettiqute and use common sense but is it safe to assume if I don't see a sign pictures are ok?
Are there any good places to go for a run in Tokyo or Kyoto? I assume most run at parks but figured I'd check around :)

Thanks for all the help you guys have given me!

Hakone has lots of baths, which are reachable in a day trip from Tokyo:
http://www.japan-guide.com/e/e5209.html

Running around the Imperial Palace is very popular. Yoyogi Park is more pleasant if you don't like to partially run besides busy roads.