If you have equal LHCP and RHCP from an antenna at the same time it is called linear polarization. You can have LHCP or RHCP at different times from a single feed if you include a switch, on a patch for instance.
If I got LHCP gain is 12.43 dBic and RHCP gain is 2.43 dBic from an array with single Port excitation, then what will be the array so called? Is it LHCP array?
What should be the minimum difference in between RHCP and LHCP gain of an antenna or array?
I think that would be a LHCP array, but 10 dB RHCP reject is not very good. The ratio depends on what you want to do. If there is a strong RHCP interfering signal then 10 dB will not be good enough, but if all you want is to pick up most of a LHCP signal then 10 dB is fine. Some uses require 30 dB or more cross-polar reject.
Yes, you can have both LHCP and RHCP from one port, this is a reconfigurable antenna that can switch between LHCP & RHCP with the help of a switch(a diode, MEMS etc).
For the difference in gain as Malcolm said it depends on your application, but I think 15-20dB is mostly good.