If you're BP is in the tank, the hottest spot he could get to is IN the tank. So, why would you put the thermometer outside of the tank? Some people put it outside, some people put it inside. I personally put it inside.
The bamboo is fine, the cage isn't a high one and there's no sharp edges for him to fall on. He should be fine.
Also, just so you're prepared, get a spray bottle for misting. You won't need it during the winter most likely, but in the summer you'll use it a few times. The advice about the plexiglass or aluminum foil on the screen is very good, I had to do that to my 20g.