Zip up hammock

That sounds profoundly uncomfortable. All the water vapor your body/breath gives off would be trapped in there with you and you'd be damp if not wet by morning. I would never hammock like that.

I only use a tarp if it's actually raining, and that makes it possible to sit up and read, eat, boil water, etc. All of the things I like to do when it's raining.

This pretty much can't work for the reason the above poster noted. You will be soaked from condensation alone. Also, as someone who has spent many nights in a hammock with a winter top cover (basically what you describe but not water proof. It is breathable and used to keep warmth in the winter) it is far more confining and much less out in the open than simply being under a tarp. Plus with a decent sized tarp and not sideways blowing rain you can set it up in porch mode and still have great views.

Every once in a while, someone pops up at Hammock Forums "hey, I want to make a waterproof hammock" – and folks who have tried come out of the woodwork.

You might go check out the multiplicity of threads on the subject there. If you can get the search function to cooperate.

You could try to make a tarp like this:

http://www.backpackinglight.com/cgi-bin/backpackinglight/forums/thread_display.html?forum_thread_id=50566

+1 on the condensation being a problem. Some use top covers with a breathing hole in bitter cold and they gather plenty of frost. Do that in wet humid weather and you'll be swimming by morning.

The trick is to use a tarp and put the view side up in "porch mode," propped up with trekking poles or found sticks, or tied off to handy branches. With luck, the view is in one direction and the weather is coming from the other. Raise and lower as weather and daylight dictate.

One of the coolest things with hammocks is to have your breakfast while sitting in your hammock and the rain going pitter-patter on the roof and watching the raindrops make rings in the lake at your doorstep. After you've had your second cuppa in dry comfort, you can load your pack while protected and finally stash the tarp after you have your rain gear on.

It just doesn't get any better— unless a deer comes to drink or a loon calls :)