If you cut players during the draft,

a) When do they leave the team?
b) If they don't leave until after the draft ends, do they still count as active players during the draft?

There's no roster moves during the draft (other than your drafted players arriving on your roster) - players you cut will leave your team at the next free agency stage, and players you offer contracts to will evaluate them at the next free agency stage.

When you queue a player to be cut, he is automatically removed from your active roster and you can't put him back on your active roster unless you cancel the cut. If you're on the depth chart page, you'll notice that they don't have the '+' button to activate them. They will also be listed in the "pending contracts" area (My Team->Pending Contracts)

So basically the scenario is I've got a number of cuts arranged to make room for my draftees.

As they won't leave till free agency, does that mean the draftees never make it onto the roster after the 60 limit is achieved?

You can actually be over the limit through all of free agency - you don't have to cut to 60 players or be under the salary cap until the training camp stage.

Ah, kk, I can relax a bit more now then. ;)