Hi

I'm totally new to this game and I'm a little unsure what happens after the taxation phase,

Do you start totally a fresh with the cards that you have played in the last round burning them all or do they stay active for the next round and then you go back to phase 1 then add to these cards in play repeating the 7 phases?

Also in challenges do you get power points for a millatery challenge that is won? Or do you only get this if the challenge is unopposed?

All cards in play stay in play until something makes them leave. You just start a new turn with the Plot phase.

You do not claim power for your House by winning challenges (winning a power challenge makes you take power from your opponent's House card), only for unopposed challenges (and winning dominance and effects that say you do).