yes Jane is correct re licence..... if you intend to hire any vehicle make sure you have the paper part supplied with the new style licence as well. You may not be asked for it by some companies, but some will.
Both helmets and seatbelts are compulsory under EU law.... Greece is part of EU so same laws apply even though sometimes ignored here in Corfu
Re kids and scooters/quads ... the most common problem recently has been parents who hire them and then let their younger children ride them. We have seen many tourists injured in accidents involving scooters etc and this year we witnessed one fatal accident involving a quad..... so be warned
Not sure on minimum ages... we have asked several times but never got a clear answer. One thing we do know though is that unless you have a full m/cycle licence you are only allowed by law to hire a 50cc scooter or quad. Some companies will hire you any size, but be warned... like all EU countries, failing to have the correct licence does invalidate any insurance and we all know that insurance companies will try to get out of settling any claims if at all possible.
One final thought... if you did have an accident and you were not 100% legal you would probably find that your holiday insurance company would refuse to cover you as well. This could mean that not only would you end up having to pay for any damage to hire vehicle but you could also end up with very large medical bills as well.