Hi Mike, welcome to the site.
I have an area outside of the shower, where my Hardibacker meets my sheetrock on a vertical wall. This area will not get any moisture. How do I tape and mud the two boards together? Joint compond and paper tape? Mesh and thinset? Paper and thinset?
Short answer, you don't. Nothing is needed here as they should not be attached.
Also, in the shower that has the hardibacker walls, do I mesh tape and thinset the inside corners on the walls and what about where the boards connect to the ceiling? do you tape and thinset the boards to the ceiling as well? Let me know, thank you.
Yes, mesh/thinset the inside corners and also the joints of the hardibacker to hardibacker. Do not tape or thinset anything to the ceiling. Just use a caulk joint between the tile and drywall.