I'm not too concerned about keeping it hermetically sealed from bugs. The house as originally built had a deck on the back that ran the full width og the back of the house from the kitchen window past doors to the living area to the back of the master bedroom, where there are additional doors. Some time later, the porch with a shed roof was built on top of the deck behind the bedroom, with the cheapo screen door allowing passage from the screen porch to the deck. No changes were made to flooring in the porch - it's just the original decking and crawly bugs can come in that way. Flying bugs could get in a little easier with the barn door, and I figured I might use some kind of felt to close it up if I can get the door to hang close enough. But we don't have a problem here with mosquitos, which is welcome because both my wife and I are natives of New Orleans, and the mosquito is the state bird of Louisiana. The pestiest flying bugs here are brown marmorated stink bugs and asian lady beetles, both of which are non-native to the US. They find a way inside the main house. Everyone here gets them. If I can't keep them out of the house, I'll never keep them out of the porch.
The barn door hardware comes with a guide for the bottom of the door to keep it from swinging. This is needed even without the doggie door. We get some crazy winds sometimes. The hardware kits I've seen have different anti-swing options. The one I plan to use mounts on the floor and runs in a groove routed in the bottom of the door. This hides the guide.