Thanks for your questions!
1) I would say (and play) that if enemy avoided with invisibility cap is defeated, it's not longer haunting the location when you'll come back. You can explain it fiction-wise by assuming that opponent left because it was bored by the lack of a victim. Also, the reason for such ruling is that you shouldn't get double-points for one enemy in case you would return and defeat it "again" in a fight. Of course, you may make your game run harder by assuming that avoided enemy stays on the location, ready for a fight when you'll return, and gives no additional points after being defetead for the second time :)
2) you have to spend points to get an advancement - do it as character is advancing, like paying for something :)