Is this black spot on my back serious? (photo)

It normally takes 1-2 weeks for spots to heal after electrocautery treatment.  It's important to use sun protection to optimize healing.

Since the mole on your upper lip is raised, I would recommend a shave excision.  This should remove the mole so that it is not protruding anymore, but it is possible that there still could be some pigment remaining.  If the pigment still bothers you, then you would need to have it...

All moles should be excised and set for pathological examination. After freezing the mole, the freezing would destroy and no proper diagnosis is possible.See a plastic surgeon for excision of the area and hopefully there is enough pigment left that can be examined and hope for the best