No, you don't have to live in base, in fact living in base can be bad for you. You will have "Gateway" travel AS LONG AS you live more than 130 miles from your base and you live in the lower 48 or Alaska. (I think it's 130, or something close. Hawaii, you're on your own.) The company will provide transportation from your gateway airport to your base to start your trip. The Gateway airport has to have X number of flights a day but there are plenty. The down side is since this is a benifit the company provides you, under some IRS reg, the value of the ticket is considered inputed income and you have income tax withheld against the cost of the ticket. So you didn't ever want to go to your base.
Same thing about hotels in base. If you do get reserve in your base and don't live there, the company will provide a hotel but the cost of the hotel counts as income and you have tax withheld.
You'll probably never see your base in the middle of a line because under Flag regs, when you hit your base, you get doubleout rest. You're down twice the number of hours you've flown since you were last in your base and the company won't want to loose you for 3 or 4 days in the middle of your pattern.
A lot of JFK lines don't really start in JFK. The first leg may be a deadhead to ORD. So the line will pay deadhead pay JFK-ORD but if you live in ATL, the company will just buy you a ticket ATL-ORD.