Do plants with non-green leaves have chlorophyll and photosynthesis?
Yes, it is possible, but not necessarily the case.

Non-green leaves with chlorophyl: There are leaves that don't appear green, but do have chlorophyl and therefore can conduct photosynthesis. (See, for instance, refraction effects in white caladiums or the link in the answer by Resonating).

Non-green leaves without chlorophyl: There are leaves that don't appear green because they have no chlorophyl, and so do not conduct photosynthesis. (See, for instance, parasitic plants like the Oronbanche linked by Cactuswoman or this Monotropa)