Reverser,
Hot start:
it all depends....
If the ground crew includes a techie (more and more seldom), the comm would be something like
"Ground - Flight deck, hot start engine no 1" with instructions procedures from the book.
Under normal circumstances, no techie, the comm would be:
"errrh"
"Flightdeck - ground, Yes?"
"errh, I think, emmm, smoke and stuff, I mean ground to Flightdeck, I think that sometning went terribly wrong"
[Flightdeck follows procedures for hot start]
Hydraulic failure:
"Flightdeck to ground, stop push, hydraulics low"
"Ground to Flightdeck, push stopped, awaiting instructions (our coffee is getting cold)"
Decompression:
Decompression will not normaly occur under pushback or while still in contact with pushback crew. You would have to have one hell of a slope on the taxiway....
These days, most ground crew consists of a loader with limited xtra training, so don't expect extensive knowledge of aircraft specific problems. They have been informed about how to handle emergencies but haven't had them drilled.
As I have never flown any aircraft with cabin crew, I don't know the comm with them. Unless a flight with my gf counts: "Do you have to fly so high". "Its called altitude, dear, its for our safety". "Well, I don't like it. You want coffee?". 10 mins later: "When do we land - I have to pee".
/Flap Sup