I recommend you see if your Deere dealer can get you an x749 that would be an ideal mower for your lawn. I would go for the 62" deck and mow the lawn at about four inches height. I live in the mountains in New Mexico and find this works well on out steep slopes. The 4ws is much better for turning uphill and the K92 transaxle setup with 4x4 can hold up to the steep hills you have.
From the pictures, you have, what 2 or 3 acres to actually mow? Or do you mow the whole place?
I'd agree with jenkinsph that you need something way more than a riding lawnmower for these hills. Also, that's not a finished lawn or anything close to it on those hills... more like a closely mowed field, and it looks like it's also got a fair amount of potholes, ruts and other unevenness to deal with. That JD model or a subcompact (I know, I know, these are expensive options), would handle that kind of rough service and the 4wd would be safer and not as likely to scar the ground so erosion can make more ruts.
You might want to look for a decent used subcompact or JD X type. Kind of rare, but you might get lucky and find something affordable. You might also keep an eye out for something like a used Steiner or Ventrac set up for hill mowing. Again, not cheap, but perfect for your place.
Beyond that, if you're physically fit, one of those walk-behind landscaper mowers, maybe 60", would probably do the job, but I sure wouldn't want to be doing that every week.
Maybe the mowing was a factor in why the place was up for sale. But it sure is pretty! Hope you don't get banished to the outhouse too often!