Can partially collapsed lungs repair themselves?

Yes. A partially collapsed lung is due to an air leak from the inside of the lung through the covering of the lung, out into the space between the lung and the ribs. The site of the leak has some damage, which will heal itself later. If only a very small amount of air leaked into the chest cavity, that air might go away (reabsorbed by the body) without treatment, and the lung will re-inflate.
Yes. Yes they can, it all depends on the cause of the collapse, the symptoms and the level of collapse. Unfortunately after a colapse and repair the chances of another collapse on the same lung is pretty high. It is important to determine the reason for the collapse. You must see your doctor to determine the reason and discuss your treatment options.