The answer has to start with the question of which God you're asking about. Certainly the Greek Gods did; Apollo was heartbroken when his pursuit caused Daphne to turn herself into a laurel tree, which is why he wore (and we still crown winners with) a crown of laurel. However, if you are talking about the traditional Abrahamic God, the one worshipped by Jews, Christians, and Muslims, then the answer would be the God knows what it feels like (He's omniscient, and knows everything including what's in our hearts which generally contain a lot of guilt and remorse), but that He Himself does not feel them. Since He knows everything and is infallible, He cannot make mistakes (is inerrant), and therefore has nothing over which to feel guilt or remorse. He can change His mind (see Genesis 18, in which Abraham argues God down over how many righteous people would save Sodom), and He certainly feels regret over our falling short of what we might be, but in traditional Western religion He does not feel either guilt or remorse. Reform movements have a range of different visions of God (within the three religions under discussion) and in some of those God does feel remorse. But a full discussion of them lies outside this exegesis

Here is the good news. God is make believe so if you like make up a god that feels guilt and remorse, or make up one that does not feel guilt or remorse. Take your pick.For that matter make up two gods at once, or a hundred gods one to suite your every mood and you can make them all vanish in a blik of an eye.