If anything, the conditioner will attract/keep more dust if it is uncovered. Wipe down with a wet cloth after every ride, and depending on your climate, condition as necessary. I live in a pretty dry climate an only oil my dressage saddle about once/month at the most. It is virtually impossible to keep a saddle clean, especially if it is buffalo or other coarsely grained leather.
No. Just because it's wiped with a damp rag does not mean it has had all the natural oils sucked out.
I wipe down dusty saddles with a well rung rag, and my bridles get rinsed lightly and wiped dry with a well rung rag after each ride to get the sweat off. I condition every 10th use with a light conditioner. I do a super duty conditioner (like lederbalsam or hydrophane) twice a year.
I buy the microfiber cloths from the 99 Cents Only store, but you can get them anywhere. They suck up the dust, and I use them on my boots after every ride.
I don't think there's anything wrong with applying conditioner every time. I use Effol Lederbalsam once a month liberally, and in between that I use the same rag to wipe my saddle down. It keeps it nice and supple, and is really quick and easy.