They do make bullnose tiles for edges that have a rounded edge. If the tile that you are using does not have edge pieces that you can get, you could pick an accent tile to border it that has bullnose tiles, use a decorative tile molding (pencil, dental, etc.), caulk the edge, or lightly sand and stain the edge to match the rest of the tile (last resort). We do that with travertine in the showers (not sure exactly how the tile guy does that).

If the metal face of the fireplace gets hot, using a construction adhesive or organic tile adhesive would most likely be a problem. Attach wire lath and a mortar bed or cement backerboard then set the tile in thin-set mortar. Alternatively, use three pieces of 3/4" thick granite with clips that connect the pieces together and to the metal fireplace or the adjacent wall framing. The clips usually slip into slots cut into the unexposed edges of the granite. The granite solution is easier and better looking in my opinion. Be sure to make the top piece larger than the side pieces especially if the metal fireplace is typical and much wider than it is tall.

Not a pro (and I’d hire one to do this!) but ripping out and starting over will produce a better result. The tile on the floor should be level to the wood. Looks like it’s lower now. Also adding metal trim schluter to all exposed edges would finish it off. Also Consider pulling the interior wood trim (which they tried to cut around) off and replacing it post install.

Not sure about the Durock or wire mesh, which is one of your main questions. I don't do tile work, but I do design fireplace covers and gas fires and often help people with an overall finished look. So I created these two mockups to help you along the way, seeing as we don't know what direction you are really going with the style of decor in this room. I was going to create a mockup also showing a more traditional style mantel surround, but ran out of time. If you go that route, those two electrical supplies will likely need to be moved. Tile floor to ceiling with glass doors and burning glass with geoshapes Tile half way up with glass doors and matching metal angle iron trim to hide the exposed tile edge. We can make that metal trim any depth necessary. This works well if you want a rather contemporary look. You can also use wood molding. (Shown with gas burning fireballs.)