So which one is it? Or all (any one) of them? Or just a further-away and less convenient parking location especially for gaijin?
All of them. Btw, the maps are a little deceptive. The overflow lot next to the station is really just across the street from the hotel's main parking lots.
And do they make any mention how to park there (first need to go to hotel, stop in front of it on the street, to get some parking pass/ticket, or park first (at which location?) and then just walk over to hotel and get the parking pass for the free/included parking there, or pay for parking and they reimburse it then, or..?)?
It doesn't mention the procedure, just that parking is free. I would try the main lots first, and if they are full just double park to check in. Then check with the staff about the procedure for their overflow lots. Mostly likely they just validate your parking pass when you leave.
Also, the hotel mentions that they will hold your reservation until 10PM so be sure to call them if you plan to arrive later than that.
And on an unrelated question: are the ETC tolls payable in coins only, or both coins and paper money? Is exact change needed (machine only to drop coins in, for example), or do they provide change (human cashier is there)?
You can pay for tolls in cash, coins, or credit card on most of the major national highways. Smaller private toll roads may only accept cash. Exact change is rarely needed.
Btw, ETC is an electronic pay system that requires a special credit card. Don't use the toll booth lanes marked ETC or it will be an embarrassing experience.
And how much are the ETC tolls typically (e.g. for Fukuoka Airport to Beppu, how much should I expect it to be?)?
Depends on the distance driven. Fukuoka Airport to the Beppu area is about 4100 yen.