IMO, crypts leaves staying close to the ground is more the exception rather than the rule. It seems like I much more frequently see them growing more like yours. I do recall one other thread talking about crypt leaves laying lower if they are 'out in the open'. I observe this a bit in my tank where the crowded side of my Crypt wendtii grows more upward, while the side toward the open middle lays down more. But, I wouldn't really say that it is laying on the substrate.
As to cutting the leaves, I don't believe it would cause any damage. Though, I'm not sure the new growth would come in much differently.