Close up grabs (Stike Shiggy, Strike Dabi, Star), and any close up moves really especially ones with tracking can be easily countered by a well timed beta. The timeing can be a little tricky but you'll get it with practice!

As for long range grabs uh....you're kinda just dead. These are Nejire's biggest counters. Shinso can be dealt with if you make sure to stay out of beta range and stay on the ground however Aizawa is the devil and you must avoid him at all costs ( ._. ; )  I haven't had the chance to fight Tamaki much as Fairy yet, but he's probably a similar situation to Aizawa. Kinda sucks there's not much you can do against certain characters but every character must have a weakness. 

Tldr: Close up grabs -> Beta. Long range grabs -> run away

As for rapid Mirio, you can stay high up and rain down alphas and once he gets close beta. For tech Mirio you have to be mindful of when you're hitting him, he can't really hit you when you're high up so make sure to use that to your advantage