These are the results of my web search for you.
1.It is possible to bring a pet into Turkey with its own passport. However, on returning to the UK you would be subject to quarantine laws.
2.There are no problems, with bringing pets with you both as long as you obtain a "pet passport" and have the proof of all the vaccinations.
Also,
My sister moved two cats from US to Turkey and there was no problem. I think airline charges might be different from UK but from here was around $25-30 for each cat. I can find out more details from her later. For now that is all I can find.
Good luck. Felix will love Turkey.
