He was attempting to stop Combustion Man to keep them alive, to prove tht he wants to join them, yes.
No, Sokka was just trying a new angle of attack (pun detected *beep beep* pun detected), not doing so to rescue Zuko. At that point, they thought he was flung over the cliff, but the screenwriters rescued with the old "hanging tree branch over cliffside for the good guy to grab instead of falling to their death" trick.
That was "okay, bribing the damn merc (whom I hired to begin with, but we won't go into that) to go away and forget that this ever happened isn't working. Since this means they now have to kill the merc because the bastard will not quit, the least I can do is help."