1) I'm showing perfect ADS-B coverage but, presumably, this is from my transponder rather than PAW? If you have ADSB Out through a Mode S transponder then, yes, it will use the transponder antenna. If you're talking about ADSB In (which you are when considering Vector) then, no, PAW won't use the transponder antenna but the ADSB antenna plugged into your PAW. ADSB are much stronger signals than PAW etc, so are easier to receive. If you only have the room for one external antenna then make it the PAW antenna (Tx/Rx) and leave the ADSB antenna (Rx only) where it is.
2) However 'ModeS' option shows nothing which makes me question above?ModeS doesn't show anything on Vector or the ATOM station for anyone. I think it's a facility that was included in the design of the app but not yet used.
3) If I go down the external aerial route, I would need a ground plane, and presumably, a different one to that which my transponder uses. Could I use a separate ground plane for both PAW aerials? Yes, you would need a separate one to the transponder because you want to distance the antennas as far as you're able from the relatively powerful transponder transmission. The two PAW antennas can use separate ground planes. One is for the PAW Tx/Rx and one is for ADSB Rx only. You want to get the ADSB Rx away from the transponder antenna because they're both using 1090 MHz and the transponder can swamp the ADSB Rx. They don't need to be massive distances (i've heard 20 cm bandied around) but the further the better. As a very, very rough example, I have PAW under the bum of the left seat of a Eurofox, the transponder under the right knee (ish) and the ADSB In under the bum of the right seat. They all have their own aluminium ground planes.