In order to get a comp pick for an unsigned first rounder, do you have to offer him his demands, or will him just being unsigned trigger the comp pick?

You must make the initial offer. He must then reject it. 

Thanks, and that offer must be at the price he's demanding or can it be at a lower amount?

No lower than what he's demanding.

Thanks!

I've had an offer rejected but it was countered.  My original offer was his intial demand and was counted as today.  I'd have no problem signing him for that $$, but with a 58 Health that's too much of a risk.  If I withdraw the offer on a Blue X rejected offer vs the Red X refused offer to gain the prospect $$ back, do I lose that Type D?

If your first offer was rejected, and he comes back with a counter offer, you are under no obligation to make a new offer on his updated demand.  You get your type D by doing nothing at that point.

Are you saying that you made a second offer, and are considering withdrawing it?  If so, I'm not sure what happens then with respect to the type D.