The word "fault" means a defect or an imperfection. God has no defects or imperfections, and it is nonsensical to suggest that the contrary could be true, for if God had defects or imperfections, they would not be recognized as defects or imperfections by anyone, including Him, and thus to call them defects or imperfections would be utterly meaningless.
Further, if God were anything other than perfect-in-every-way, nothing could be trusted at all since He is the source of everything. If nothing can be trusted, then knowledge itself is impossible.
So, no, our sins are not His "fault".
However, God certainly does cause people to sin, and we see plenty of evidence of this in the scriptures. The key point here is that it is not a "fault" for Him to do so.
According to the Bible, God caused the murder of His Son, Jesus (Acts 4:27-28). God caused the Israelites to stray from His ways (Isaiah 63:17). God caused marauding Sabeans and Chaldeans to murder Job's children and servants (Job 1:21-22). God caused King David to take a census that He then condemned Him for taking (1 Chronicles 21:1-8; 2 Samuel 24:1). God caused Eli's sons to disobey, dishonor, and disrespect both their father and God Himself (1 Samuel 2:25).
So, you may be wondering, how is this not a "fault"?
The logic is pretty simple, really:
Premise 1 - God did these things.
Premise 2 - It is impossible for God to have "faults" (defects or imperfections).
Conclusion - Therefore, it is not a defect or imperfection for God to do these things.
Frequently, we like to think that the rules that God has given us apply to Him as well, but there is no real reason to think that, and every reason to reject such a concept. It might very well be a "fault" for me to cause someone to sin (Matthew 18:6), but that does not mean it is a "fault" for God to do so. After all, He is God (Psalm 135:6). Does not the potter have a right over the clay (Romans 9:21)?
Having explained this, God calls on all men everywhere to repent (Acts 17:30). And as the scriptures clearly explain, He will hold you personally responsible for all of your sin unless you call on Christ for forgiveness (John 3:18).
Do we have free will?
Yes. They could only exist if they were the product of his deliberate design. Moreover, all the effects of sin could only exist if God designed the universe so as to produce those effects, thus they must also be his fault.
Take just one example: lust. Lust is the basis for many sins. If we did not experience lust, we would not commit any of its related sins.
Science has proven that lust is hardwired into the brain: it exists there as a fundamental component of the actual design of the human brain and body. So if God designed that brain and body, he must have consciously and deliberately programmed it to experience and be motivated by lust. Yet there is no credible reason for any intelligent designer to program our brains to feel lust. Lust serves no intelligent purpose. We don't need it to experience love. We love friends, neighbors, and family without experiencing lust; even love for our spouses isn't produced by lust, but respect and admiration and compassion. And we don't need lust for procreation. Most life on earth reproduces without even brains, much less the means to experience lust. God could have designed the universe so that love between a man and a woman spontaneously generates a child.
However, evolution by natural selection (in other words, the complete absence of a compassionate and intelligent designer) explains why our brains are programmed to feel lust. Indeed, it is the only plausible explanation of that fact. As a product of God's design, it makes no sense at all. But as such a product, God shares responsibility for all the sins motivated by the lust that he installed in our brains.
The Science of Lust