I do not think so, I believe you need a minimum of 6 months validity on your passport in order to travel. Best to check the airline but i am pretty sure you need a renewal.

Presuming you are a British passport holder, your passport is valid for any EU country, which includes Cyprus, until expiry. The airline has nothing whatsoever to do with the matter.

Not always strictly true. - some airlines do make it a condition of travel that you have x amount of months left before expiry so best to check the individual airlines T&Cs. Given that you can have up to 9 months added to the new passport the easiest thing for peace of mind might be to just renew it now before you go.

Yes you can. If you're a British passport holder your passport is valid for travel within the EU, right up to it's expiry date. It only has to be valid for the proposed duration of your stay. It is not a decision made by the airline, and they are not within their rights to deny you boarding (unless they want to end up paying IDB compensation and getting some bad publicity).

I remember EZY getting themselves some rather unfavourable publicity because they did just that, and it was to Cyprus. The eventually have to admit that the check in agent got it wrong!