1. I think I am calling a turn and value betting/pushing the river. No-one but no-one minraises with a queen here - if they do have it they either slowplay, or re-raise bigger to push out the flush draw - and unless they have AA or 33 you're way ahead (of 66-TT probably).

4. Pretty much read dependent. If an aggro player plays like this I check; if it's a passive player who'd play Axs or Qx like this, I value bet and call a push. I think the only hands that beat you and check this river are 33 and 55 held by someone who's unreasonably scared of the straight possibility.