Is it possible to have several lists on your board but customize which lists are visible? E.g. TODO, DOING, DONE, VERIFIED where everyone not on QA team could customize their view to have VERIFIED column hidden.
No. But see How do I hide and show entire tasks lists in Trello? , which has
".. there is a feature request called Collapsible Lists which you can support by e-mailing email@example.com."