How to fill gap between jagged side rock fireplace & sheetrock?

Is this a working Fireplace? It would need to be Fireproof ....

You can mask off the Rock and use insulation foam, fill the void and let it dry, this stuff is messy, then cut it back to flat and smooth it off with sparkle, prime and paint

Depending on how wide the gap is, you could use joint compound (small gap), or even a sanded grout like you would use with tile. Choose a color that blends well with your rock...

It would depend on how big the gap is. If it is small enough, you might be able to create a "patch" with mesh to bridge between the two materials. If it's deep, you could fill it first. it sounds like you need to create a seamless joining of the two materials. If it's large, you might need something more.

I would caulk it and carefully paint the same color as your wall.

can you fit small stones? otherwise, maybe spackling or putty?

If it is bigger than a 1/4 inch, I would use a foam backer rod, and then use paintable caulk over top.

Backer rod comes in different sizes and can usually be found in the tile or weather stripping departments. It will help you use less caulk and the bead doesn't have to be very wide. It also provides a good base for the caulk.

http://www.homedepot.com/p/M-D-Building-Products-1-2-in-x-20-ft-Caulk-Backer-Rod-71480/202066515

Caulk would be a good fix, paint it to match the wall