Are there car rental companies on Gozo?
Is it allowed to move with rental car from Gozo to Malta and back by ferry?
Yes, just use google and you'll find them. Same with Gozo ferry - it will tell you if cars are allowed on it and how much it costs.
We recommend renting through Auto Europe (www.autoeurope.com) and it looks like Avis has a location there. You could compare rates directly on Avis' website versus Auto Europe, which is a consolidator and typically cheaper.
I would like to refer you Sicily Tour Services. They have fine packages for your trips anywhere in Europe.
http://www.sicilytourservice.com/
You do know that they drive on the left as in the United Kingdom, Ireland, and other countries with a British history.
Malta is small, the roads are narrow and there are many cars and buses, so traffic can be very slow and driving may be very stressful.