Lady stoneheart says Dominance: Put Lady Stoneheart into play from your dead pile. Then, discared all power from your House card

If i kill her to satisfy claim or deadly in the Challenge phase can i bring her back in the Dominance phase even if i do not have any power on my house card or do I need at least 1 power on my house card in order for her ability to work?

You can bring her back even if you have no power in your House. The "discard all power" effect applies only is Lady Stoneheart came back from the dead pile successfully, not the other way round.

And that's a good general rule to point out. When two parts of an ability are linked by the word "then," the requirement is that the first must successfully resolve for the second part to resolve. If the first part is somehow cancelled or impossible, the second part doesn't go off. Contrariwise, if the first part can resolve, it will, regardless of any concerns of the second part.

KristoffStark said:

This is important because if the second part has a separate cost, target, or play restriction (think Wendamyr's 2-influence cost after the "then" or Meera's choice of a card to blank after the "then"), you don't pay, choose, or consider it until the first part is confirmed as successful.

Fair enough.