There's a different di2 plug that can be installed in individual parts for error checking but if that's working off the bike it's most likely wrong installation. Download the Shimano manual and follow the instructions it by the letter, it's usually pretty good.
Not unless it's actually acting up and need to be warrantied. It's really simple to install and most reputable shops will be fully booked for weeks.
What mode it's in? What happens if you try synchro shift?
There's a different di2 plug that can be installed in individual parts for error checking but if that's working off the bike it's most likely wrong installation. Download the Shimano manual and follow the instructions it by the letter, it's usually pretty good.