Does the bladder heal itself over time?

Yes. Our bodies as a whole usually do a great job of self-repair; as long as there is nothing to hinder the work efforts. Examples include poor nutrition, certain disease processes, certain exogenous substances and medications.
No. Unfortunately, the only organ know to rejuvenate itself is the liver, if the damage is not too severe or constant. 3 ways to keep bladder healthy: drink 6-8 glasses of water per day, not sodas, alcohol, or coffee. Avoid tobacco, either direct or secondhand. Learn kegel exercise.
Depends.... It depends on what it is healing from! from a urinary tract (bladder) infection - yes. From cancer - no. We need a little more information on what you're healing from. Or go see your urologist for answers specific to your case.