I suggest getting the nearby radio tower, it will increase it over time.

I have most of them, how much does it increase standing?

And, all I need do is capture it, right? Nothing beyond that?

yeah just capture it, it aint fast but it helps. Ofcause getting missions to the cities can give you a nice boost but really dont hit civis :)

I believe the standing drop you are getting for killing NATO/gendarmes is a bug. They usually do not cause a standing penalty like civilians but in one of my games it started doing that but stopped after I restarted and restored from save.

Oh that has happened constantly since I have been playing, I thought that was a normal part of the process, I cause stability to go down, which pisses everyone off at me

When you shoot bluefor you lose standing. Thats not a bug.
Police stations and radio towers are your main way to raise standing. Also killing Crims.