KeyForge: Call of the Archons

Yes — during Step 1, if you have enough Æmber to pay the current forging cost you must forge one key (only one key can be forged in Step 1 each turn). (page 4)

You may choose any house shown on your identity card or any house for which you currently control a card; you cannot choose a house unless one of those conditions is met. (page 4)

No — during Step 3 you may only play, discard, or use cards that belong to the active house unless a card ability explicitly allows otherwise. (page 5)

Yes — an "Omni:" ability may be triggered on any ready card you control even if that card does not belong to the active house; using it exhausts the card. (pages 6–7)

When creatures fight they simultaneously deal damage equal to their power, then prevention/ward/armor are applied and remaining damage is placed as tokens; if a creature has damage equal to or greater than its power it is tagged for destruction and later put into its owner's discard pile. (pages 7, 33)

No — if both creatures are destroyed simultaneously they are not in play to have their "Destroyed:" abilities (that require them to be in play) resolve afterwards, so such abilities do not trigger; destroyed effects are handled at the destroyed timing described in the timing chart. (page 30; timing: page 33)