Just found out that when I look at Style Library (root site) with SPD and
Properties, on secutity tab, the permissions list/page/view is ALL blank.

When I do the same on another site-collection, on secutity tab, the permissions
list/page/view is FULL of "restricted reads."

Did all security settings get erased when I added a folder to hold my XSLT?
If so, do I just added it back as it is on another site-collection?

I would begin by creating a test user account. I would copy the Full Control
permission set to create a custom permission  set, assign the test user this
permission set, and begin to take away their access  until you figure out what
access they can have and what access they can't have. I assume you already
aren't inheriting permissions from the parent so I won't go there...

Did you publish your style  library? and approve the style sheets?

No I did not "publish" the Style Library, and I do not know what "approve"
means for a file. After all, the Style Library is only a container for files.

This "publish" stuff has me totally confused. To me there are only read  &
write file types, and saved and not saved files.

Thus, as I make changes with SPD, I ONLY save them. I have not "published"
anything. I have not had problems (yet) with this process.

The problem  is that the Style library  is a document library, and not just a
container for files. Document libraries have additional capabilities that
control access  to and the visibility of files stored in them. Things like
whether require approval, versioning, etc are turned on will affect the
visibility of files.