There are many national and international funding bodies for arts and animation. I would recommend researching some of those for production support funding.
Pretty much. You submit an application for funding and they,(at their discretion) may or may not grant you funding to complete a production, but you still own it.
It is best to read everything first to check eligibility and the likelihood of success. Eurimages, for example grants around 90% of its available funding to co-productions, so if you have a project and can collaborate with a producer in another country it might be worth applying there. It can be as much work securing funding as making the film in the first place.