After I start the drops for pink eye, when will my eye stop being pink?

It takes a while. Pink eye is most commonly caused by a virus and does not respond to medication (including eye drops). Bacterial eye infections are responsive to medication. The good news is that the infection usually improves within 1 week or so, at which time the redness should go away. Depending on the severity of the infection, it may take up to 2 weeks or longer for the eye to look completely normal.
It varies. "pink" eye is caused by a virus and it is usually self limited. Most ophthalmologists will treat their patients with either a combination steroid and antibiotic drop or a antibiotic drop alone. Usually the steroid component is what "quiets" the eye and makes the "redness" resolve quicker.
1-2 weeks. It can take a while depending pn the cause of the infection. Most pink eye is viral and will clear up in 1-2 weeks whether or not you use any drops.