The weight of a male betta's tail fin is actually very heavy for them because they are so large, so the tail drooping is not uncommon and nothing to be worried about. If you carried around a 75-lb back pack all the time, wouldn't you let your shoulders slouch on occasion? It's like that.
The white spot could be ich, can't really say without seeing a picture. 'Course, 'white spot' is pretty vague, could be a fungal infection too.