Yes you can. Motherboards are transferrable from one MacBook of the same model/year to another. Another option is to just transfer the Firmware chip by desoldering and soldering. It is fairly easy, the chip is an 8-pin sitting next to the fan vent cooling grill.

I agree, there is no need to remove all internal components, including the motherboard. Get the glass and LCD off, then just lift the motherboard slightly in the battery connector to slide the old battery out once you have the adhesive off, and likewise slide the new battery in, connector first.

Personally, I use the warming drawer on my stove to heat it up, works nicely.