I have a 12-hour stopover in Guangzhou Baiyun Airport and want to get a 72-hour visa.

Well, you don't need to apply for any visa. You can enjoy the 24 hours visa free policy because your stopover is just 12 hours. But if you wanna go out of the airport, you need to apply for the Stay Permit when you arriving at the airport.
About the luggage, you don't need to recheck them if the two airlines have luggage check through service. Thus you are advised to consult your airlines and confirm this information. :)