I always check it with the luggage. We have a special car seat bag that we got when the girls are infants and I can actually fit both girls booster seats (highback) in one bag if I put them in just right and I can actually stuff our coats in their as well. Its wonderful! I never gate check them as I hate having to carry all that extra stuff through the airport. I have never had an issue and we travel 1-3 times a year (depending on the year).
Re: Flying w/booster seat
Lurker here. We fly at lot.
I check mine. The airlines have always put them in a bag for me and I have never been charged for them.
Mother of two wonderful boys! Blessed through adoption.
We gate check our backless boosters. Back when DD was still in the nautilus we checked it with our luggage.