I wouldn't try removing the frame. It appears to extend in both directions and may damage a larger section of wall.
You are right to be concerned about taping compound or spackle sticking to a metal edge. I would recommend making a patch that is larger than the opening by about an inch on each side. Then trim away the back paper and the core of the patch so that the part being put into the wall is the exact size of the hole. That will leave you with a "flange" of paper facing on hte room side that will be easier to tape and compound to blend it in.
Your general approach to support is right, but you may not need a brace between the studs.
To support the patch, cut three pieces of 1x2 wood that are about 6 inches longer than the height of the opening. Position each piece of wood vertically in the cavity, one near the left side of the hole, one in the middle, and one near the right. To attach each strip pull it forward against the back of the plaster or plasterboard. Hold it in place by screwing through the face of the wall into the top and bottom sections of the wood strip. You can screw the plasterboard patch to these strips for support.
Finish the attachment with tape and taping compound. I would probably use mesh tape on a patch of this size and support.
Before you attempt to repair it, make sure that the cause of the ice dam has been fixed. You may need to insulate the ceiling, and/or add a styrofoam vent baffle to the inside of the roof to prevent that area from getting too warm. Ice and water shield installed on the roof probably would have also prevented this from happening.
The repair will depend on how long the wall was wet and the extent of the damage. From what I am seeing, you should be able to fix this damage without needing to pull the whole wall apart.
Let the wall completely dry out and then check for signs of mold growth. A dehumidifier is not going to dry out the wall for you. You will also need fans aimed at the wall to circulate the air. Industrial blowers will do the best job, but a household box fan on high will also do.
Mold may be on the surface of the plaster, and it could also be under your trim. Use white vinegar to clean off the areas affected by mold and then use a mold killing primer such as Killz on all of the damaged areas.
Once the wall has been primed, then you can repair the plaster. You will need to get a bag of plaster of paris and a 5 gallon bucket of drywall joint compound. Mix the plaster with water per the instructions and mix in joint compound in a 50/50 ratio. Trowel on the mixture and blend the repair into the existing wall. The advantage of mixing the two is that you can use thicker coats of plaster without having it crack. It will also dry much faster and shrink less. The joint compound will make the repaired area harder and will bond better with the wall. Over time plaster dries out and becomes brittle. The gypsum in the joint compound will help prevent the plaster from deteriorating. In the old days, they would mix in chopped up animal hair (usually horse hair) to strengthen it instead.
You may need to apply more than one coat. You should also sand lightly between coats and feather the repair into the wall as best as you can. Any imperfections will be easily seen once the wall has been painted.
Once you are satisfied with the repair, then you can prime the wall. You can use any kind of primer that is compatible with the color coat. I prefer using primer that is rated for interior and exterior use since it is less likely to peel when you put the final coat on.
Repaint the wall and trim. If you can, repaint the entire wall at once so there isn't any differences in the sheen. It may look different if you only paint the damaged part.
Once the wall is painted, you shouldn't be able to tell that there was ever any damage there.
Best Answer
A patch will be virtually undetectable if done properly. Professional plasterers do often skim the entire wall but there is no reason to texture the entire wall if you do not want to.
Using texture is an easy way to hide imperfections, so is a great idea if you (or your contractor) do not want to spend the time and effort (and $$; time = $$) to patch it properly. Call a different "drywall guy", preferably a reputable professional plaster company and one that doesn't leave unscrewed floating corners like in top pic.