As the others have said. You aren't an object. There's no reason you couldn't get a crown, circlet, tiara, or headband and cast it on that. At least that way you can take it off when you don't want to be noticed or a distraction (or trying to sleep without a blindfold for the rest of your life). Sure, a hood or cowl would hide your flaming head, but that would work on a flaming crown or circlet too.
No. The target is "one object touched". You are a creature, and generally speaking specific body parts aren't targetable unless the spell explicitly calls it out.
You could, however, get a friend to kill you, then cast it on your lifeless body.
Crazy idea: what if you cut your own hand off, had someone cast continual flame on it, then cast regenerate to re-attach it. Would it still be on fire?