I took the injection...3 month contraception. Instead of birth control pill. Could losing hair be a side effect?

Yes. Yes, but it is not a permanent loss. Each hair goes through a growth phase, a "resting" phase, and then falls out to start the cycle over again. Conditions like pregnancy and use of hormones synchronize much of your hair so that although they will all fall out at the same time they will then start growing back.
Doubtfull. Though I have heard that happening from a few patients, i'm not sure it is actually caused by the depoprovera, but is usually caused by something else. For example, many women start the shot after having had a baby, and it is common to lose hair in the first months after having had a baby.