Sorry for delayed response:
There's as much room under a Hennessey hammock as you care to make. It's determined by the type of fly you take and the height you sling the hammock.
Staying inside one for a day, or more, is easy for me. There are way more comfortable ways to lie down, sit, stretch &c in supported positions than in a tent. At least for me. I cannot speak for others. Though I do wish to point out, ahem, that experience and experimentation with an actual hammock is a better basis for an opinion than speculation.
Cooking, as others have mentioned, is easy under or beside the hammock.
As too weight, I don't know how light tents can be now. I guess with modern materials and designs the answer might be "very", I haven't looked at buying a tent for years. When I was in the market for a new tent or hammock, nothing came close to a hammock for lightness, and a Hennessey hammock was *for me* a great combination of flexibility, durability, lightness, and comfort. With extra long supporting ropes and broader slings than the standard Hennessey, together with insulating foam, my hammock weighs about 2.5 lbs. It's much better made than anything *I* could whip up with a machine, and more convenient, comfortable, and consistent in performance--not to mention insect-proof--than the improvised hammocks I previously made at each campsite with cord, nylon sheet, and found sticks, as much fun as that was. But they may not be for everyone: my son could never get used to sleeping in one, and prefers a tent. As I said: all shelters, like any designed object, involve some sort of compromise.