If the God they worship, the violent one who gives virgins to murderers, doesn't actually exist. Do prayers somehow migrate to the One True God, or are they in a sense simply worship intangible idols? (or worse real demons)?
According to the CCC, its the same God though worshiped wrongly, and sinfully.
According to my Orthodox Priest, they don't really worship the same God.
Well, that's the thing.
In their minds they are worshipping a deity who simultaneously (1) created the world and (2) gives virgins to murderers.
In reality, God certainly has attribute (1) and certainly does not have attribute (2).
We both know that there is no God other than God and we both know that Muslims intend to worship God under his true attribute of (1) while they also insist on assigning to Him the false attribute of (2).
The way to look at the situation with the maximum amount of Christian charity is to see Muslims as people trying to worship God correctly but who have been demonically deceived into worshipping him incorrectly.
According to my Orthodox Priest, they don't really worship the same God.
I'm not going to reject his view out of hand, but I would say that they claim to worship the God of Abraham.
From the level of evangelization, I believe the best way to win Muslim souls for Christ is to fasten on the common belief that God is the God of Abraham and to explain how the Gospel is the true and final account of God's self-revelation to mankind and how the Koran cannot be.
I would say that though they profess to believe in One God that they don't "KNOW" the God that they profess to worship. In strictly human terms, if I know of you and for some reason I think you are a mean, vindictive, hateful person and you're not, then it is my perception that is wrong. You are still you, and I know you exist but I don't know you as you truly exist. So God is God, and he remains God no matter how wrongly someone perceives Him.