We just got back from plane trip with our 7 month old. We bought an inexpensive convertible that we will leave with my in-laws. I would gate check your carseat. DH worked for a car rental company a long time ago and I would NOT use one of their's, like ever.
I wore baby when I was checking our bags, then had to take him out to go through security. I wore him onto the plane, but again had to unbuckle my carrier for takeoff (funny though out of 4 legs, only 2 times was I asked to take him out). but yes, you can just hold baby the whole time :0)
I wouldn't hold my breath on an empty seat, they are kind of non-existent these days. I have flown 9 legs in the past 3 weeks and there wasn't an empty seat in one...they were asking people to go later! Crazy.
You'll do great!
SAHM and Wife to Matthew
Forever Mommy to Thomas