Free will is the capacity of an individual to make his/her own choices and either to act or not act on those decisions. It really has nothing to do with God outside of theology and philosophy of religion.
This is a philosophical question, and as such, can't really be answered experimentally. At the risk of sounding sophistical and clintonian, it depends on what you mean by "'really' free". You can be controlled by environmental factors and still exercise free will by some definitions of that term.
UPDATE: A brief summary of compatibalism: http://www.rep.routledge.com/article/V014SECT1
According to them dogs do not have free will but people do. Probably talking about ethics and moral side of "free will" question, not physical or religious. But even there "free will" is just an illusion in your mind.