No you can only use the extra dice against an undead bane. To tell if they are undead is by looking at there traits in the top left hand side of the card in bold. If it says undead then yes you can use that power.
Think of her as the cleric she is....wielded of the holy light purging the undead.
Example of undead trait card - ancient skeleton henchmen
Can I use her power to add a d8 to defeat a bane that doesn't have the undead trait?
No.
And secondly, when are undead really dead? When you defeat them? Or should the bane card explicitly state that they will resurrect so you need that magic trait?
I assume by "dead" you mean "defeated", and "resurrect" you mean "undefeated". They are defeated when you succeed in the Check to Defeat like any other monster, per normal game rules. Only if/when there is some text in the Powers section that says otherwise would it be undefeated.
There is no special rule regarding undead (or any other type of) monster. If neither the card nor the rules say there is some special defeat condition, then it doesn't.