Well this is the perfect forum for such a question!
Firstly, don't worry they'll be fine, the Captain is notifed that he has live animals on board, including their location on the plane, and he'll make sure the temperature is ok.
As for the logistics of it, if you're travelling on the same flight it is both easier and cheaper. You basically take them as excess luggage and pay the charge by weight. Once you've decided which airline you're flying with, call them or visit their website to find out how they operate with animals. Most have a dedicated cargo number, and someone can tell you the procedure and requirements. You should be able to 'lock' the cage by using a plastic tie wrap that can only be cut once fitted. Don't think a padlock would be a good idea.
You can use an animal shipper to facilitate all this, or with a straightforward direct flight like this you could do it yourself. A vet can tell you the vaccination requirements, some have to be done within a certain time of flying. Watch out for hidden charges at the other end (customs/immigration/vet). Ask the airline how to organise a vet to meet the flight in cyprus if necessary.
And finally a handy hint- get an old tupperware box, drill holes near the top so you can tie it on the inside of the cage door with plastic tie wraps. Before the flight, fill it with water and freeze. Then it won't spill while the cage is moving around. Once in flight it should have defrosted enough to drink!
Good luck!
Moufflon