When determining initiative for adversaries most don’t come with a value for the “cool” skill so you’d just roll the green die the’d start with in presence correct? Same would be true for someone without a value in vigilance, they would just roll the starting pool for willpower?

Correct. In both cases, it'd be an unskilled roll, just as it would be for a PC who lacked any ranks in Cool and/or Vigilance.

So for Street Thugs (Presence 1 I think), you'd just roll the one green die if they needed to roll Cool for initiative.

If you had a persuasive adversary (Presence 4) that didn't have ranks in Cool, then you'd roll four green dice for initiative.

Thanks morning fire for the confirmation.