If you leave the armor in the first chest, it will stay there as long as Haven is available. I doubt it would be available if the chest wasn't looted and Haven was destroyed, though, so if you want to eventually sell it, do so before In Your Heart Shall Burn.
I don't know about the Avaar or Qunari armor, but the dragon preorder armor was only good for the inquisitor, and scaled at low level. I usually sell the dragon armor without even putting it on, because the first set of crafted armor is as good anyway. If the Avaar and Qunari armors are usable by companions, I'd suggest outfitting whoever can use nice armor and selling what you don't need. The armor and schematics you find soon after leaving the Crossroads are better anyway.