I remember reading earlier in the thread suggestions to use the smallest size flash drive you have. Maybe that was just due to the ease of formatting it--I don't know. Certainly no harm in trying your 32GB but if you can't get it to work then see if you have an old one laying around.
In my experience, the newer firmwares are definitely faster to connect than the older ones, and also fixed a bug many of us were experiencing with laggy GPS positioning. I've been on 20200307 for about a month now and it's working great. I know others (mainly GM owners I think) got audio glitches with that firmware and have had better luck with 20200306. You can freely flash between those two firmwares (or even older ones) and find the one that works best for you.
Yes, you should see if your car keeps the Carlinkit alive after powering off the vehicle. Look for the tiny LED lights. With my Ford, power to the USB ports is cut as soon as a front door is opened (after turning off the ignition) but I know some folks have ports that stay powered longer or even indefinitely.
I still occasionally have cases where my phone will not reconnect to the Carlinkit. It's only happened once in the past two weeks though, and a quick reboot of the phone gets it working consistently again. Possible that Apple improved this with the iOS 13.4 release, I dunno.