Engine is straining to pull the car - 1995 Buick Regal

By pull, what do you mean exactly

they shifted the load by pulling ..can not work any more out

Probably needs a tune up. Engine load increases on hills, putting a lot of strain on ignition system.

This is normal. The heater when turned on requires an extra strain from your engine. your engine automatically realizes the strain and adjust throttle to acomdate the change in your engine

when was the last time u changed your trans fluid, my honda was doing the same till i changed the trans fluid and now it dont wine at decelleration

More than likely is a glazed fan belt. At idle there's not alot of strain/pressure on it. When you're driving belt is pulled much harder. A glazed/smooth belt will silo over pulleys showing low charge triggering your light.

how many horses/cattle? it will pull around 7k-8k pounds with little problems over that and you will be straining transmission. this is also assuming flat land if in mountains wouls be less weight

Turn cruise control off. You should never use cruise control when pulling a heavy load. It puts more strain on the transmission and engine. 

I would think that it was more likely that some wires shorted due to the strain caused by the broken motor mount and that short circuit is what burned the clutch coil out. With the coil not energizing, the electromagnet that pulls the clutch together and engages it, will not function. The pulley is turning, but the compressor isn't. The end result is if the compressor is not turning, no AC.

your flex plate inside your trans is broke