There are, but most are not new-build ones.
Certainly, but this isn't one.
It is and it isn't. Cambridge itself is a city where a car is an encumbrance, but the rural hinterland is one where it's an essential, similar to Oxford and the reason why both have pretty good P&R provision. With South being on the southern edge of the city, it does seem a missed opportunity to remove park-and-ride-to-London traffic coming from the south from the roads into "City" station and provide for them with a dedicated access. A few might use Trumpington P&R for that and go to City by bus, but hardly any will.
It's not relevant for park-and-ride-to Cambridge traffic, though, as Trumpington handles that.
I suppose, that said, there is Whittlesford Parkway. Maybe the view was that the likely outcome of a car park would be more cars in Cambridge as people switched from that? And unlike Oxford most of the collection of villages to the south of Cambridge have their own station on one line or the other, while the string of stationless villages along the A14 and A428 would more easily reach "City" or North.