a wrote:

Is it possible to become an ATP-rated airline pilot without going through
the military and without owning your own plane? Do people do this often?

Yes and yes.

What would the certificate/rating and career paths be?

Typically, the pilot studies for the private certificate, then the instrument rating,
then gets a CFI ticket. That lets him or her work for an existence (can't really call
it a living) while building hours. The commercial ticket and multi-engine rating are
picked up during these years. The next step usually is to some sort of charter
operation or light freight (like check hauling), but some people I know have gone
straight from CFI to feeder airlines. Making the transition from piston to turbine is
tricky. Getting into the right seat of a feeder airline is the next step.

I recall flying the cattle section of a Fokker into Knoxville once. The captain
invited any interested parties to check out the cockpit (this was well before 911).
When I mentioned that I owned a Cessna 150, he said "Yeah, I've got a lot of time in
those!"

There are other ways, as well. Several colleges and academies will get you into an
entry level airline job.

George Patterson
This marriage is off to a shaky start. The groom just asked the band to
play "Your cheatin' heart", and the bride just requested "Don't come home
a'drinkin' with lovin' on your mind".