Blowdown settling drum water is typically sour water. We send ours to our sour water tanks for treatment in our various sour water strippers. I would think that it would be a very bad idea to pump this water directly to an open tank. You could add a slipstream of the sour water to your quench water during the quench, similar to sludge injection, but you would still need somewhere to store it until needed.
As far as open top versus roofed, it’s dependent upon your situation and your uses. We have an open top tank. We maintain our level in the tank with stripped sour water, so no H2S, with a utility water backup if we lose our stripped sour water makeup. One of the benefits of an open tank is that you can quickly get a visual idea on the status of your tank. Plus, when it comes time to clean out the fines, I imagine an open top may be easier in that you could wash it down with a firehose from up top. The open top helps the heat dissipate a little faster than a roofed tank would.
Not everything by any means, but it may give you a starting point.