Correct, you'll be good. Sorry I use speech to text and I'm I'm running around trying to answer lol. Best of luck to you.

It will tell you if it's grounded to the panel or I very least to a ground rod. Which is where you want to be grounded anyways..

Keep in mind when working after someone else that especially when mistakes are made that they may have never grounded it all the way back to the main panel or any sub panels.

Yes what you are saying is fine and safe if they didn't do some weird ass backwards other things. If all they did was skip grounding the outlets then pigtailing it will work perfectly fine as well as going to a larger wire size is fine.