What is the difference between one who mentions the name of Allah while committing a sin, without seeking His help, and one who mentions it while seeking His help? And does one who mentions it seeking His help to commit the sin, or one who merely deems it permissible, become an infidel? And is only he who deems it permissible to mention Allah during the commission of a sin, without seeking His help for it, sinful?

A person who mentions the name of Allah while seeking His help to commit a sin has committed something forbidden. He does not commit disbelief unless he considers it permissible (halal (definition)) or treats it lightly. This is the difference. As for someone who mentions Allah Almighty during a sin without intending to seek His help or blessing, and without treating Allah's name lightly, he does not commit disbelief by that and may not even incur sin. He will be rewarded for mentioning Allah, even if he is heedless, because occupying the tongue with obedience is better than silence or occupying it with sin.