Red dot on lip permanent for months. sometimes turns purple, no puss or raised bump though. what could this be? melanoma?

Dot : A red or purple area on the lip may represent a blood vessel called a venous lake. See a dermatologist for a proper diagnosis and treatment if needed. Melanoma may a times be brown/black, black, dark purple or even reddish in color so get it checked to be safe.

Possibly venous lake: Red or purple spots on lips are often dilated veins, most commonly from trauma or sun damage. These are more common in middle-aged or elderly people. Any new spot that is not going away should be shown to your dermatologist to make sure it is not a problem.