The accusative on “ĉevalo” means that she wasn’t on the horse in the beginning of the action. In other words, she wasn’t on the horse and he put (through lifting) her onto it.
To compare, “Li levis ŝin sur la ĉevalo” means that she was already on the horse in the beginning of the action, and was still on it after the action. This could mean that they were both sitting on the horse, but he helped her stand up on it.
I would never interpret it as going “to air”, unless the context implies that jumping on a horse is still considered to be on the horse; which would be strange to me.
Hoping it helped.
I don't quite agree with your second interpretation. I would argue that the location relates to the subject, since he is performing the action. I picture Li levis ŝin sur la ĉevalo as follows: he sits on the horse (and she possibly as well) and he lifts her into the air, while he remains on the horse. I don't quite see why sur la ĉevalo should relate to the object of the sentence, like you're describing.
Good spot! It actually can be both (but yours will be more often without context here). The precision “sur la ĉevalo” in principle applies to the action, but in this case there are two places for the action: the new place for her, and the place for him. I agree that without any context saying otherwise, the meaning where he stands on the horse since the beginning is a good interpretation :-)
I was thinking of the grammatical point pictured in “La birdo flugas en la ĉambro” in the following link when I initially wrote my comment: http://bertilow.com/pmeg/gramatiko/rolmontriloj/n/direkto.html#i-sgp