I found the solution in case anyone is wondering in the future. All I had to do was turn "Physics" off in the object settings of the airplane. Very easy fix.
I had a similar issue when I first used custom paths. The plane “popping” off the ground usually means the spline points aren’t smooth, or one of them is set at the wrong height. It helps to check the path from the side view and make sure the slope is even and not dipping underground. The falling effect at the start of the loop can also happen if the starting speed or position resets too suddenly. I’m still learning too, but I’ve seen some simple tips about this on a small site called flying411, and they mostly suggest checking the path points and start settings first.