stephlac -- Let me try to answer some of your questions based on my (somewhat limited) personal experience. I'm sure that others will jump in and cover the gaps in my explanation.
First, when you redeem HH points for an airline award (or a VIP award that includes tickets), you receive a voucher to present to the airline. You don't receive airline miles in your account (the way Marriott does the transfer). Then you have to exchange those vouchers for actual tickets. Depending on the airline, some of those vouchers are for an award ticket (e.g., UA, in V class), some are for a "revenue" ticket (e.g., AA, in N class). Not completely sure if technically you can truly call it a revenue ticket, but AA tickets issued on a HH award earn mileage.
One thing to remember, that inventory for tickets issued on, let's say, award ticket on UA paid for with HH points, is different from UA Mileage Plus award inventory (typically, availability is much better with HH awards). And yes, there are blackout dates, but it has been reported (
http://www.flyertalk.com/forum/Forum57/HTML/003845.html ) that sometimes those blackout dates while stated by HH, are not enforced by UA. An important thing to keep in mind that those tickets (at least on UA) are not upgradeable.