I note in Heb 11:8, 9, we have the following inspired commentary:
By faith Abraham, when called to go to a place he would later receive as his inheritance, obeyed and went, without knowing where he
was going. By faith he dwelt in the promised land as a stranger in
a foreign country. He lived in tents, as did Isaac and Jacob, who were
heirs with him of the same promise.
Thus, the act of Abraham in leaving his home country to travel afar was definitely an act of faith! This answers the second question.
The first question is more difficult - how did Abraham recognize the voice of God and thus know that it was not a false God or false spirit? - We are not told so we do not know for sure.
However, we do know that on this and one other occasion (there were probably many more) Abraham recognized the voice of God. The other famous occasion was when God told him to sacrifice his son, Isaac (Gen 22).
All that can be said, is that as a prophet of God, Abraham knew the voice of God when he heard it.