Hey there!
Well, that is quite a few requirements ^_^ Such a buggy/pushchair/world-facing stroller definitely exists, even if, of course, it will come at a price (and with some compromises, perhaps).
I will start with the air-filled wheels which, for bad terrain, can definitely help. Better than that is, however, their size and/or stroller suspension. Be sure to rather pick a quality stroller then a cheap one which you'll feel getting looser and wobblier every time you go in harsher terrain. My first thought was actually the Gesslein S4 which, along with larger (not inflatable, though - harder from, really quality one) wheels sports a very nice shock absorbing ability. I know, I know, your partner wanted inflatable wheels, so I will also suggest a Valco Snap Trend 4 Sport (no suspension but nice large Sport air-filled wheels, and a fold as you wanted, too), the Espiro Sonic Air - nice & robust, with a large hood and inflatable wheels) or the Joie Mytrax - the tires of the Mytrax wheels are filled with foam but are very close to the inflatable wheels. The Valco Baby buggy, the Sonic Air, and the Mytrax models have an adjustable handle along with that.
Lastly, as you don't mention the budget, I will also mention three-wheelers from Baby Jogger - the City Mini GT2 or the City Elite 2 have solid wheels with no punctures while they can go over ANYTHING. Fathers especially like them as they are sporty and durable. I know they fold outwards, but at least they very flat when folded.