Your right, in my opinion keeping a razor in the bathroom is a BIG no no. Another big no no is getting water near the pivot pin when rinsing the blade. Water will get down between the scales and blade and it doesnt dry quickly. This could explain the rust at the pivot you mention.
I would try the least aggressive option first with the rust and that would be rubbing a little metal polish on it and seeing if that will remove it.
Be careful though if your blade has any gold wash or similar on the blade metal polish is likely to remove it.
If you could post a picture that would help us assess the damage
Hope that helps
Grant