I would use all purpose joint compound (green lid) and paper tape if you wanted a relatively simple and reliable patch. Follow with more thin coats of all purpose until you're satisfied that the lump is fared quite wide. (Lots of thin coats is always better that putting it on thick and then having to sand, because as you probably know, drywall dust gets everywhere.)
For a more complicated but quicker fix (saves time taping, saves a day of drying), use 6" wide mesh tape and setting compound (marked on the bag/box as "20" or "45" setting type joint compound). Apply topping mud right after doing all the taping. Repeat topping mud as needed. On the downside, mixing your own mud from powder is a bit of a pain, and it goes hard before you know it. (I say topping mud because it sands easier than either AP or setting compounds. The downside is that it's frequently only available in large-ish quantities, so you'll probably have to feel bad about tossing some out.)
It would help on the textured spots to
sand scrape down the texture around where the tape is going, as that will be less of a lump for you to fix later.
Your first pic looks like it has a considerable dimple at the screw at the top of the picture. Make sure that's flat before you tape.
In any case, texture after -- there's no point in trying to texture before.
I forgot about the possibility of ceiling texture containing traces of asbestos... so, at the very least, scrape instead of sand, spritz with water to contain dust, and use a decent p100 mask. Or have it tested and know for sure what's there.