Do mosquitoes have feet

No. The leg of a mosquito has multiple parts, the most distal of which is known as the tarsus. The end of the tarsus contains a "claw" which is a pad of hairy, adhesive organs (tenent setae) that allow the mosquito (and other insects like flies) to attach to smooth surfaces.

The mosquito does not have a "foot", which is defined as "the terminal portion of a limb which bears weight and allows locomotion". The setae of the mosquito don't qualify.