Generally the contraceptive effect of Depo-provera will kick in within two weeks of the injection or within 24 hrs if the injection was received during the first five days of your period.
Each injection gives protection against pregnancy which will last about twelve weeks. Yes it can be used at at an interval longer than 3 months ; ie on a “ad-hoc” basis.
Once you are menopausal ( have not had a period for more than 12 months), the likelihood of natural conception is very low and you would not require contraception at that point.