The main advantage to the 4S is that if you move past this build - you will likely want 4S (or bigger). I think it's been played out enough that 3S works for some builds, but most people, if they stick with the hobby, quickly move past 3S to 4S. So it's more of an investment thing.
It also provides higher voltage, so the capability of the machine is extended. If you decide to add weight, r move to a heavier machine altogether, you won't be stuck.
The larger props (if power correctly) will give you more lift due to more surface area to move air. It's not quite as simple as 1 is more stable. I have had smaller props react more stable in heavy wind - because they are cutting the air more often, and not being affected by the external forces as much. But obviously the trade off is that they are spinning faster/more to provide the lift - which could translate to vibrations on the craft. Larger props spin less, may create less vibes (if balanced of course) and may translate into "more stable." But they will be more subject to wind gusts, etc.
Pitch will be less an issue. It will act like more/less scoop to the air. There is a formula I'm sure for how much air it will move changing the pitch. Unfortunately I don't know if offhand.
Hope that makes sense. Very non-scientific
