Is it normal for your insurance Co to ask you to delete your name from the check to the finance Co for damages?

generally no

did the ins co cross out your name or did they just put the finance companies name on the check? If they crossed it out they just committed a crime- its called altering an instrument(check) . If they left your name off the check , the claim was for a total loss and you owed more that the insurance settlement , the finance company would be entitled to the full settlement from the insurance company . If that's not the case your name should be on the check too.