There are no trip or duty rigs, just flight time and dead head time, which pays 1 for 3.
The company will position you so that you get rest before you start flying. All patterns start and end in your base. So yes, if you start flying at 6z on day one, you'll have to travel without pay on your day off. Same thing at the end of a line, if you get into JFK at 2200, you'll be going home on your first day off. But if day one is just a limo ride JFK to DOV to be in position to fly on day two, they could ticket you ORD-PHL with a limo to DOV but you'd only be paid JFK-DOV.