Both chetsinger1 and jmauldinuu do not have a "connection" - you have two separate flights.
So, as Frank II said, you cannot remain airside. You will have to go through immigration and customs in FCO (unless you are stopping in another Schengen country on your way to FCO), then make your way to your Vueling gate, passing through security on the way. Customs is just a walk through, unless you are "randomly selected for additional screening" (rare but possible). But immigration and security wait times cannot be predicted. At least you won't have to wait for your checked luggage to arrive at FCO.
Yes, if Vueling makes you check your bag, they will charge you for it. If you prepay on their website, it's much cheaper than if they make you pay at the airport. As everyone says, you cannot predict how strict they will be, so it's best to err on the side of assuming they will follow the rules exactly. Do get your boarding pass in advance (should be able to do this before you leave the US). And as said above, they are strict with their times; if you miss the deadlines, they won't let you on, but will want you to buy a new, last minute priced ticket.
If everything is smooth and quick at FCO, you'll be fine. If it isn't, you won't. How lucky do you feel?
What I did in a similar situation (FCO to PMO on a separate ticket, the same day I was arriving at FCO from the US) was buy two tickets - one three hours after my arrival, and one ten hours after. I did this because I was looking at last minute ticket prices on that route for my arrival day (a Monday), and they were starting to skyrocket; two advance tickets were still cheaper. That way, I knew I'd use one and throw away the other one, and I just counted the total price of the two tickets as the cost of getting from FCO to PMO. In the end, I made my 3 hour flight with time to spare, and I even had to pick up checked luggage and recheck it - but of course, I could just as easily have missed it.