What you're talking about is not a second officer in the US, it's a relief pilot. From what I understand, as the relief pilot, you can log the time in the seat that you are flying as SIC.
As for you other questions about Jetblue and Southwest, I know that Southwest requires PIC turbine. I'm not sure about Jetblue. You can have a million SIC hours, but unless you meet the PIC requirements that those airlines have, it won't do you a lot of good.
I don't know what foreign airline you're looking into, and, frankly, it doesn't really matter. The important question would be: What is the upgrade time to captain? You might be able to make FO there in a year, but would it be better to look into a domestic regional where the upgrade time is shorter? That may help you get to your goal faster than going foreign.
The most important thing, at least domestically, is turbine PIC. Based on your goals, you need to go wherever you can get that soonest.