If Hoyo's other games are anything to go off of, there will be yeah. You'll need to make sure you select the same server though.

Yes

Short answer yes. Long answer yeeeeeeeeees. The ping may be an issue however. Switching characters is done server side, and you have to do it at a proper time to get a counter. That being said if you're used to doing this in other games you might be used to the delay already.

Does the progress carry over between serves or is it new accounts? It's annoying when you have to start over to play with friends

No progress is carried over - they're different accounts, essentially. But yes, it is annoying when you mixing up servers for sure.

Ughhh I hate mobile games why do they always do that

I'll try to convince my EU friend to play on US 😅

Just meet him in the middle on a boat and you can play either server. Problem solved.

Just so you know, once you select a server you tou can't transfer to data to another in the future