a full-hopped back air does auto cancel but the hit box only stays out until a bit after you reach the peak of your jump, so if your enemy is still at ground level it most likely wont hit them. Since the lag can be punished, I usually only use back air as a situational edge guard, or use a full hopped back air to hit the opponent while there on platforms.
Neutral air (the same applies for down air aswell) DOES NOT auto-cancel from it being full hopped. It only makes it so that every kick comes out, but there will be lag on the landing. But a good strategy that I use so that there is no lag upon landing, is that I do a ground jump and immediatly after a mid-air jump (similar to DJC) and then do a nair. Although it sends you a bit higher than a full-hopped Nair, every hit comes out and theres zero lag. It's basically just a safer way of doing it. It also puts you in a good hieght to hit your opponent as he jumps up off the ledge, and if he gets by it, it will be hard to punish because there wont be any lag. But sometimes if you do it to quick only a short hop will come out and make things worse, so its really whatever you prefer.
Up air auto cancels off of a short-hop and forward air auto-cancels of of a full-hop.