Barcelona airport transfer

One ticket to Barcelona and a separate ticket on Vueling? Or one ticket all the way to Split. There is a huge difference in processing between the two situations so which is it?

Arriving in Barcelona nonstop from the US means you will have through passport control.

This site may provide help

https://www.aeropuertobarcelona-elprat.com/ingl/flight-connections-barcelona.htm

Two separate tickets

You have to clear passport control in Barcelona (EDIT: I thought Croatia was in Schengen zone by now, apparently not so, so I think you'd only be checked in Croatia); I don't know Barcelona airport well enough to know if you also need to clear security.
Have your Vueling boarding pass printed in advance, and know that if you are late arriving on American, you will lose that Vueling ticket if you miss the flight. 2.45 hours looks feasible, but it gets quite risky if the American flight is delayed by more than 45 minutes.

It's my experience that in Europe you always have to go through a security check between flights, yes, even for connecting passengers who remain airside. Will you already have your Vueling boarding passes when you land at BCN or will you have to check in for the flight? I believe Croatia is not yet in the Schengen zone, so I think you will have to go through passport control there. I have no idea if you would need to go through passport control in Spain.

It's my experience that in Europe you always have to go through a
security check between flights, yes, even for connecting passengers
who remain airside.

Sometimes you do, sometimes you don't. Other EU-countries are considered safe as the security checks there are trusted, so passengers arriving from other EU-countries do not have to go through security again. In addition there is a short list of non-EU countries where the security checks are considered to be equivalent to EU standards, one of the countries on that list is the United states, so passengers from the US do not have to go through security.

However, not all airport can separate arriving passengers from trusted non-Schengen countries and not trusted non-Schengen countries, and at those airports all transfer passengers arriving from non-Schengen countries have to through security. And as always there are exceptions to the rule, the UK has opted out of this so if you have a transfer in the UK, you always have to go through security when arriving from an airport outside the UK.