I visited several nice onsen in Beppu :
Hoyoland (保養ランド) -
http://www.secret-japan.com/onsen/show.php?&selcode=106
A really interesting onsen, which is more of a mud bath! You really have 20 cm of mud at the bottom of the huge baths found there... Mixed outside baths and nice old inside baths.
Kaihin Sunaba (海浜砂湯) -
http://www.secret-japan.com/onsen/show.php?&selcode=107
A specialty of Beppu: sand bath with a view on the sea. Again, well-worth a visit if you are there!
Nabeyama no Yu (鍋山の湯) -
http://www.secret-japan.com/onsen/show.php?&selcode=108
Looking for a wild onsen? This one is really small - so not that great - but you have to take a bus and then hike a little to find it! Print a map to have a chance to locate the bath!! Unlike the others, this one is free. There is another one nearby... but I could not find it.
For exchanging your flat while in Beppu, I am sure that some Japanese would be interested. But I have no idea how to find someone for that, sorry...