If any faith that we have is God's then why did Jesus says "Have faith and doubt not"...if man has no ability anyway of his own...
Jesus said, "If you have faith..." That implies nothing about the source of faith. But, you quoted mid-sentence, "Have faith..." That implies having faith is our choice, as if Jesus is instructing us to get faith, rather than just telling us what we can accomplish with faith.
I've been in Pentecostal churches were their false pastors would explain the lack of successful faith healing is a result of a lack of faith of the congregation. You can't choose to have faith that someone will be healed. I can't say, "I need to put away my skepticism and belief that the pastor is a fraud, and have faith that God will heal through him." It's not within my power to make that choice. (Besides, it's not biblical that miracles depend on the faith of 3rd parties.)
I do not believe anyone makes a choice to have faith. But, I believe we can choose to lean on our faith, which is what I believe Jesus means when he says, "If you have faith and do not doubt..." I take that to mean, don't doubt your faith.
And why are we told to "choose this day whom we will serve" if we have no choice but have "false faith" as you said?
"Choose this day whom you will serve" was said to people who claimed to have faith in God, but worshiped pagan gods. The people were being called to stop being hypocrites and to follow the faith they already have.