The best way to add new stud framing above fireplace and tie it into existing framing

I agree with @DA01 that it'll basically be impossible to fish horizontally through studs and insulation, without having to cut open all of the sheetrock. Then you have to deal with sealing it all back up air-tight, and patching the drywall and repainting (at least the entire wall, and maybe the entire room, depending the paint match).

So a couple alternatives:

Use a wiremold type product to surface mount the conduit:

There are lots of different brands, and it can be paintable. With planning, you can minimize where you see it.

Build up the wall using cabinets or a false wall, so you can run wire in behind, on top of the existing wall.

Could be as simple as just a drywall section sticking out, or some built-in cabinets or some brick work. Cabinets of course have the benefit of being able to build in other components such as your A/V gear and speakers.

The studs are likely following a standard layout such as 16 inches on center (16 oc) this is so that plywood/sheetrock will typically fall on a stud without the need to cut the 8 or 12 foot piece. For your application, I would recommend (based on similar jobs in the past):

1. Inserting 2 x 4 blocks horizontally in between the vertical studs at the edge of the sheetrock and the edge of the insert It looks as if three would be more than enough. Secure with framing nails or screws (screws would be practical in this application). This will provide vertical anchor points for the vertical section of your wood surround.

2. Mark on the sheetrock the location of the studs for future location purposes

3. At this point you can secure the surround to the framing, or you can attach the backerboard for the tile to the framing and then attach the surround to the backer board. In this case, attaching the surround directly to the framing may give a cleaner final look. Typically surrounds are made to attach to a ledger board (basically a piece of wood secured to the framing), and the surround then rests on the ledger and is secured with trim nails, etc.

4. Attach your backer board for the tile. (dura rock etc.) Be sure to use screws about every 6 inces in the studs. You want the backer to be as secure as possible in order to prevent future cracking.

Here is a link that shows a similar project, take a look at the ledger. in most cases, guys will use a simple 2 x 4 for the ledger (Note, I have no affiliation with the link).

I would recommend taping the seams of the backer board with fiber tape, and and filling the tape with thinset when attaching the tile.

I hope this helps!