Typical fireplace construction.
The wood mantel above the masonry is usually added as the last part of the esthetics of the fireplace. Wood cannot be used as a structural element in a fireplace. This could possibly be removed with reckless abandon.
The area of the firebox, not the brick beyond the face of the wall where your insert is, usually that's where the structure is, both above and below. Don't mess with it unless you really know what you are into.
The red brick that is in front of it all TYPICALLY is just a veneer. It is USUALLY never built with the fireplace, always after, it is only for looks. It just as well be tile, natural stone or hardened mud pies.
As you expect to do, after you pop the wood off, no need to support it temporarily, then start to remove the brick with chisel and hammer, protect the floor if you are going to repair it, no need to repair more than you have to.
The biggest problem you should have is the hearth. It will be no more difficult to remove the brick than the rest of it, but the slab that supports may be in the way still. Then again maybe not, since you have a tile floor and it all looks like it is laid starting at the same level. You will need to figure out what to do the bare spot where the tile is cut around the hearth that is no longer there.
Do get a pro to remove the gas line before you take anything apart..