I am assuming that your first question is about inside corners and you want to apply the tape then apply the mud. Try applying the mud first and then setting the tape into the mud immediately after. Or purchase the sticky mesh tape and apply that to all the inside corners by first folding it in half. Then you can mud over that.
Outside corners require a metal strip with a beaded edge. The metal strips should be nailed (or screwed, nailing prefered for this since it seats better) to the corner and then mudded over.
That 3M adhesive is extremely effective. I use it on a lot of things. It would probably work but may cause other issues. Joint compound becomes very hard when it dries, I don't think you will have a problem with any sagging.