Can't help you with the terminal question - I do know that we just went to the rental car exit at the terminal we were in and walked over to the garage to the assigned car. My experience with
Budget was good until our rental in Aug. Make sure you check the car from bumper to bumper and have even the tiniest scratch/dent marked on your rental agreement or it will come back to bit you. There was a scratch on the passenger side front bumper and we unfortunately didn't report it at the time of rental. When we returned the car the agent that checked us in made such a spectacle in the garage - to the point that every other customer returning their cars heard her. I have never been treated so badly and was so embarrassed when we were done. Needless to say Budget heard from us about this and then 5 months later they actually billed us for the scratch. I sent them an email to the address listed and told them in no uncertain terms what I thought of the way we were treated and that they would now be at the bottom of our rental car list (even though they are usually cheapest) plus I questioned the entire running of the company since it took over 5 months to bill us. Needless to say they responded that the claim was miraculously closed and they hope we will rent from them again. Just be careful with even the smallest mark on the car - note everything and even take pictures to cover yourself.