I think simply saying "my mistake" or "my bad" sounds more natural than "the error is mine", which sounds a bit strange.
No. You can say dele(s) (de ele(s)) or dela(s) (de ela(s)), but not de mim. another alternative is "de minha parte".
But saying "It's Dou's mistake" instead of "It's Duo's mistake" is your mistake, right? :-)
It's essentially the same idea, one which you can also convey with "A culpa é minha" (culpa also means "guilt/blame", so it aligns better with fault in terms of intensity).