Through a fence

I have had this type of fence and removing part of the lower horizontal rail will definitely compromise the fence panel, especially the vertical pickets in that area. Perhaps moving that portion of the horizontal rail a little higher will work ok but that is a lot of work. I wouldn't worry about a "doggy door". I have found that all sorts of 4 legged critters routinely scale my 6 foot PVC fence. Fortunately, skunks are not one of them.

Is it a top and bottom stringer or 3 stringers?
My back fence is 3 horizontal supports (stringers). My side fences are 2.
Since your fence is faced on both sides, can you easily remove one side of the cedar pickets?
Removing or cutting the bottom stringer of a 2 stringer fence will affect the structural integrity. Period.
If you remove one side of cedar pickets and provide a header like in a doorway or window you will retain the integrity of the fence.
As for skunks, I have no experience.

PS: I have about 200 linear feet of cedar fence at my house, spaced 8 feet on center.

Tommy
Rio Gracie

Have you considered adding another stringer or 2 to "frame" the opening. Like putting in a header for a window or door in a building. Maybe a "out of scale" portal? I used a magnetic doggie door for access to my power shed to keep small critters out and that has worked so far. Cats have no issue climbing/jumping my 8' block wall. So far coyotes haven't dropped by but that is probably because of the big dog rather than height of fence.