So, let's say I'm farming an infinity, does it drop from Doc on regular or only true vault hunter mode? Also, for LVL 50's to drop, do I have to start a new game at LVL 50 or will they begin to drop at 50 as soon as I hit LVL 50?
The level of the drops are based on the enemies level, not yours. A level 50 enemy can drop in a range of 48-50. It sucks when you're farming a lv50 guy for hours & he drops a 48. You can get legendary drops as early as the game starts on normal mode. I've gotten a few lv4 t-fists from Flynt, & a few lv4 bonus packages from boom/bewm.
Many legendaries can be dropped at anytime by their respective bosses. However when farming them at level 50 after completing TVHM mode than you are guaranteed if a boss drops it will be at level 50. As far as I know the general statistic for chance of a drop is 1% and purely 1% chance drop. Think of it as playing slots. You could pull it 1000 times and not a hit. Next person comes along and hits the jackpot. Pure luck. even if you farm 100 times you are not absolutely guaranteed that 1% is going to happen. At least one Legendary-the class mods from Terra are guaranteed drop everytime. One random class mod will always drop. Some bosses may dropped more than one legendary piece of gear. There is a list online of who drops what but I don't remember it and list I have at home is only updated thru Scarlett DLC. It lists all Legendary gear and unique weapons. Note vending machines and chests have been known also to cough up legendary stuff but again it is at a 1% drop rate.