Will an FM1 stock cooler fit an AM3+ motherboard?

I'm fairly confident it will, though the only way to be sure is to try.

The nifty thing that AMD did was make sure that the bracket holes from AM2 Motherboards and beyond were compatible/interchangeable with one another, in terms of aftermarket coolers. I think you'd rather the stock cooler that comes with the AM3+ selection, anyway. If the stock cooler is anything like what I got with my FM2/A10-5800K (Thick aluminum square-shape with fins that radiate from a central square and a fan on top) then you'll be happy with the "upgrade." It has an actual copper baseplate, with some copper heatpipes.