How do I remove the mounting hardware of an intercom speaker, and then patch the hole?
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.