What ticket(s) do you have?
It sounds like you have one EC & Connections ticket?
The rules state that you may use "appropriate connecting trains", if you do not catch the last train with an official connection into the 2130 from King's Cross, then you do so
at your risk. If you get to Gatwick at a time later than the 'recommended' time, then you
may still get to King's Cross in time but if you do not, you are not covered as the delay was due to a flight, not the railways. Even if you get a later train that is scheduled to make it in time, if it's not a valid connection and that train gets delayed, you are at the mercy of the discretion of EC staff.
As for the specific issue of trains that are reservable (or semi-reservable, as in this case), on the connecting leg, it is a debatable issue that has been done to death on this forum. I'd rather not get into yet another debate with HairyHandedFool on this issue, so my advice is to do a search and read through previous topics.
I avoid getting into these problems myself by ensuring that when I book tickets, I try not to get reservations for the 'connecting' service by specifying a suitable via point. Of course this does then open up the other 'can of worms' about using a reservable service without a coupon, again I have no wish to debate this issue again, so see previous responses
2) Secondly, what should I do if I fail to make the connection with the train due to a late running flight? Would I be accommodated on a later service or would I be expected to purchase another ticket?
If you ask nicely, they might let you on a later service (I don't know what day you are travelling but there is a 2200 & 2330 departure M-F and a 2210 departure on Sundays). Ask at the ticket office for the ticket to be endorsed for later travel (not sure what time it closes!) if they say no, it's still worth asking the guard as it's at his/her discretion.
Officially I don't think they are obliged to let you travel due to a late flight but staff can be sympathetic to such reasons.
When I experienced a major delay on a bus due to severe congestion caused by York races, I found a sympathetic member of platform staff who endorsed my ticket. He commented that he uses the buses himself and was therefore sympathetic to my plight. I still see him around, and if I am ever in that position again he'd be the first person I'd approach!
Unfortunately at that time of night there will be less people to ask, so this may lessen the chances. On the other hand guards tend to be more sympathetic on evening services I reckon. I can't guarantee anything but you may well be 'let off'!
This well get you to somewhere not far from Lincoln, such as Newark. Obviously the railway isn't going to pay for you to get a taxi due to a delayed flight, so you may want to take out travel insurance if you don't want to pay the cost of a Newark - Lincoln taxi late at night.
The last train into Lincoln is the 2308 from Newark North Gate, which the 2130 from King's Cross connects into.
Yes. Your ticket will say how many coupons are printed and the TM can ask to see them, in which case if you're not travelling on the specified train then you have broken the T&C of AP tickets.
Not sure which question you are answering "yes" to, but it is a debateable point.
There are ways of avoiding the ticket stating how many coupons are printed, by the way.
As for whether or not a Southern guard could invalidate an EC & Connections ticket on that basis, I am unsure. I have never heard of this happening and would be very surprised if it did.
It is not as simple as saying that the terms have been broken (I'm not sure they have at all), we have had pages and pages of debate on this issue before.
As for what I would do? I'd take FCC, where ticket checks are incredibly rare, and if a check occurred, I am reasonably confident that showing a EC & Connections ticket would satisfy the ticket inspector. This would also avoid the need to change onto the tube at Victoria, assuming that the Thameslink route is open at that time of night on the day in question.