http://overthecap.com/salary-cap-space/ has the Cardinals at $9.1 million. The spotrac link has them at $4.5 million after the draft.
Going into the season with no money limits them to signing non-veteran players at league minimum who won't count against the cap (by not being in the top 51 salaries).
Pro rated money becomes dead money if a player is no longer on the roster.
That precludes the Cardinals shipping Patrick Peterson or Calais Campbell to the Vikings in return for Adrian Peterson. Any package the Cardinals could put together to free up the $15 million in cap space needed for Adrian Peterson would have to include Jared Veldheer or Mike Iupati (or both) and some more bodies. What good is having a running back if you have to gut your offensive line to get him?