If Dark Imposter exiles a double-faced card, it will only have activated abilities of the front face of that exiled card, no matter which face was up on the battlefield.

Dark Impostor gains only activated abilities. It doesn't gain triggered abilities or static abilities. Activated abilities contain a colon. They're generally written "[Cost]: [Effect]." Some keywords are activated abilities; they have colons in their reminder text.

If an activated ability of a card in exile references the card it's printed on by name, treat Dark Impostor's version of that ability as though it referenced Dark Impostor by name instead. For example, if Marrow Bats (which says "Pay 4 life: Regenerate Marrow Bats") is exiled with Dark Impostor, Dark Impostor has the ability "Pay 4 life: Regenerate Dark Impostor."

Dark Impostor can target and exile a permanent that's only a creature temporarily, like an animated land. However, because that card isn't a creature card, Dark Impostor won't have any of that card's activated abilities.

If you control more than one Dark Impostor, each will have only the activated abilities of creature cards exiled with that specific Dark Impostor.

Once Dark Impostor leaves the battlefield, it will no longer have the activated abilities of the creature cards exiled with it. If it returns to the battlefield, it will be a new Dark Impostor with no connection to those exiled cards.