The "hover" idea sounds interesting but I haven't even seen one so I can't say how well they work.

Hover mowers work like a helicopter - the blade, in addition to cutting, is a 'wing' that exerts enough lift to 'fly' the mower. Consequently, there is no resistance to moving it over rough terrain, and you swing it side-to-side on hills. Theoretically, it would travel over water just as easily, like a hover craft. They usually have a 2-cycle engine and are very light, although when its not running its dead weight with no wheels. Never having owned or used one, I have no idea how you deal with it - maybe it has a little wheeled dolly to sit on when not in use that can be rolled around or used to get it to the 'launch site'. Maybe they have wheels on them - I haven't a clue. Going by your requirements and your condition, I'd say its one of your few options. Most people will kill everything on a hill like that and mulch it or plant ivy or something.

I dont know anything about mowers but Im a 57 yr old woman who does all of her own yard work. That country grass will kick your butt. I know. LOL. I have a steep slope on my lot also. Probably not as bad as yours tho.
I turned mine into a giant rock garden. I terraced it and added some low stacked field stone walls. I planted crown vetch and daffodils and a few hardy perennials. Youll have to weed some in the beginning but its not near as hard as mowing a slope. Once the Crown Vetch takes off it crowds out everything else. Crown Vetch spreads like crazy. If you dont like Crow Vetch there are other ground covers that spread just as fast.

There are other low cost things you can do if this isnt possible. I often get free mulch/tree chippings from tree companies and the county. It takes a while to break down so dont use it on your flower beds. But its great for walkways and filling holes.
If nothing else you could just cover the whole slope in mulch. As it broke down into dirt in would fill in all the ruts and evetually you would have a smooth slope.

Ooops. sorry.By the title of the post I thought i was on match.com ...

The hover mower might work but as saxman says they are difficult to handle without the engine running. However, the newest models appear to have Honda four stroke engines which add weight and do not work well on steep slopes as they tend to flood.
The mulch might be the best bet unless you can keep a ground cover like crown vetch contained.

Get an aluminum deck Lawn Boy push mower...Very light, easy to start and can take abuse. There is a high wheel model but all these I am speaking of are used. EbAY usually has a good selection. I would look for something like that. Also a 2 cycle so the on your 30 degree hills the motor will stay lubricated.
Also Snapper & Toro had aluminum deck models as well. Again, on eBay...

Has anyone ever actually BOUGHT a lawn mower on eBay?

I've bought and sold on ebay, with mixed results. Look for something local so you can avoid shipping, because that's a terrible thing to have to deal with on mowers. For your purposes, make sure you're buying a non-self-propelled mower. Lawnboys are going to lighter than any Snapper or Toro, I think. If you find a decent one, it would be a good bet for you.

As far as other wheeled mowers, look for one with either a flathead engine (i.e., not overhead valves[OHV]), or a Honda GCV160 (which has overhead _cams_ [OHC]). These are the lighter 4-cycle mower engines still being used. They are all around 25-27lbs (engine only!), which is about 10lbs less than the big OHV engines. Actually, a staggered-deck Lawnboy with the Tecumseh flathead might still be pretty light and might survive a few seasons on that hill. If there is a non-self-propelled Honda with the Xenoy (plastic) deck, maybe it would be pretty light. I don't know of any light big-wheel mowers, but I agree that for push-only mowers the large back wheels make sense. The problem with almost all 4-cycle engines is that as they tilt too far on a slope, their oil sumps don't keep oil near the pick-up or splasher and their bearings can run dry; and/or they can get oil sloshed up to the top end of the engine which siphons into the carb or hydrolocks the engine. So there's always a chance of low engine life.

Looking at Eastman Industries, it appears they sell their 19" hover mowers with either the Suzuki 2-cycle or the Honda 4-cycle. This version of the Honda is supposed to be optimized for hillside use of up to 45 degrees according to Eastman. I was asking about this engine earlier, but nobody knows anything about it. I'd take them at their word, I guess. Anyway, the Honda has more power and only weighs 3lbs more than the Suzuki (37lbs vs 40lbs), so I think either would be a good solution for you, ann13820. If I had your constraints on mowing, I'd call them and find out if any dealers are anywhere near you. I'd also ask them how one moves the mower into a garage or shed, as discussed above. It would be a pain to have to do that with it running, but maybe you could use a low platform on casters, like a rolling platform for big potted houseplants, and just "park" the mower on that outside and then roll it inside.

Here is a link that might be useful: Eastman Industries mower site