When I get into a car that is not my own, any car, I take extra time to adjust everything...seats, mirrors, I familiarize myself with the controls, the emergency brakes, wipers and as you said, when I put it into gear and also apply brakes very carefully. I too have had that sudden jerk to a stop when applying brakes you are unfamiliar with.
Before I put it in gear, I check out the visibility and all the blind spots.
But then again, I am one of those people who always backs into a spot or pulls through, because I feel that when you are driving, warmed up and oriented, driving in reverse to back in is safer and easier than coming out cold to your car, being distracted, and backing into a busy area.
Always better to nose out and go when you aren’t “warmed up” and oriented, rather than backing out and going.
When I was learning to drive my dad said to me "parking lots are the riskiest places. Nobody knows what anyone else is going to do at any time. move carefully and look everywhere". Man, he was right on that one. Not risk of death (unless you get run over) but risk of a collision for sure.