Short answer yes, Mirage Mirror and all copying effects, copy the base creature and all the abiltys included on the card itself but not any +1/+1 counters, artifacts, or enchantments just the base ability and power and toughness of the card. So Mirage Mirror copies Necrotic Ooze base abilitys and power and toughness, thus giving it the graveyard abilitys of creatures in all graveyards just like your opponents Necrotic Ooze. Hope this helps, happy tapping:)
September 19, 2018 9:42 p.m.
The answer is yes. Mirage Mirror would copy Necrotic Ooze, meaning it has the same name, type line, power/toughness, and text box. This includes Necrotic Ooze's ability that gives it the activated abilities of all creature cards in all graveyards.
And that ability specifically says "all graveyards," not "your graveyard," so I'm not sure what you mean by "would it have their graveyard abilities or my abilities?" If you had a Llanowar Elves in your graveyard, their Necrotic Ooze would also have the Elves' ability to tap for green mana, and by extension, so would your Mirage Mirror copy.