No, this CPU does not have an integrated graphics unit. The FX series from AMD are high-performance processors with an open multiplier. These are intended for the high-end sector in combination with a graphics card. Generally, no integrated graphics unit can be operated on the AM3+ socket. If you want one, you have to use an A-series CPU (or "APU" as AMD calls it) for the FM2+ socket.

Yes, your information is correct and this is indeed possible for certain motherboards from specific manufacturers. This is more or less limited to high-end retail AM3 motherboards. But according to my informations, this is not possible with that specific motherboard you have installed in your Acer Aspire M1420. I'd contact Acer directly for further information and verification though.

comes with cooler - ready to use.