Arillas cars is probably a good bet, they will deliver car to the airport, and all his cars are in great condition.
Ditto Corfu Explore car hire if Arillas cars can't accommodate.
I know Corfu Explore definately provide baby seats, and I'm pretty sure Arillas Cars do too.
The nearest Laundrette now is in San Stef, just on the road in from Arillas, opposite Town Pizza.
Last year they charged 8euros a load, wash only for up to 5 kilos, 12 euros if you want it dried and ironed too.
5 Kilos I'm guessing is a lot of nappies, but he may charge less for a small load, would be worth asking.
Hope this helps.