1. I have been routing with default Altium settings (so not using this rule) and no problems. I believe, in case there would be something really wrong, in PCB house they may correct it, but when I asked about it, they were not very worried. I know, that when I am using Cadence Allegro, that software takes care of the angle how track is coming out of a pad.
Personally, in some very dense designs, I do not have much choice, but if you are not space restricted, maybe you can control how track goes out of the pad. Also, if you need to follow IPC standards very strictly, double check if there is nothing about it.
2. I believe, teardrops are mostly important in VIAs (I believe, teardrop helps to add some space which may be needed due drilling tolerance). But I am not expert for this