How can I improve stitching of nadir and zenith shots in 360º hemispheric panos?

When I make a pano, I use Autostitch. It allows me to be far less exact than you have been, mainly because I don't have the gear or the patience. I hold the camera in a vertical position, hold it at roughly the same place, and then rotate myself and try to keep the camera in the same position.

I shot this a few months ago coming back from Mammoth Mountain in California (9000x2300):

I used a Sigma DP2, handheld vertically, and took two rows across the entire image. I then cropped down the final result in post to get rid of any black edges.

Here are some things I've learned when shooting panos:

1. If you have a polarizer, remove it. It will do odd things across the frame.
2. Shoot in raw, if possible, and then convert all images using the same settings in post.
3. Set your camera to manual focus, infinity, and then manual exposure so that you don't get varying exposures from one side of the frame to the other. Constant aperture and shutter speed both are required here. If one square of the scene is metered differently than the others, that will mess things up in post. I tend to meter off of the brightest portion of the scene and rely on my raw processing to bring up shadows if necessary.
4. Try to avoid things that are fast moving (like crowds), otherwise you'll get odd blurs as one smaller photo transitions to the next in the larger pano. The corollary to this is to move quickly; clouds aren't static, and you have to be fast before they move on you.
5. Be somewhat distant from your subjects, if possible. You are creating what is essentially a Mercatur projection; near the edges of that projection, things will look odd, like how Greenland is far larger on a 2D map than it is on a globe. If you want the kind of warping effect that will achieve, get more local ground features in the image, if not, not so much, maybe by holding the camera higher. The plants in the pano I posted are no closer than aabout a hundred feet.
6. Avoid shooting into the sun. Lens flares will wreak havok with you (as you've seen). It will also cost you in contrast near the sun.

7. Take many many overlapping shots. You'll have to experiment with your subjects, but for the 9000 in that above shot, I believe I took about 10 images across the horizontal axis. That translates to an image every 900 pixels, smaller than the dp2's vertical resolution of 1760, so as to avoid stitch artifacts where there isn't enough information between images to get a seamless stitch. Two rows of 2640 shrank down to 2300; again, the mercatur projection effect and the fact that I was doing this handheld and therefore had a lot of rough edges to be cropped out were factors there.

   EDIT to go into more detail in specific about the zenith and nadir that I went into in the comments:

These points happen at the top and bottom of the hemisphere. If you are creating a mercator-style projection, then by the math involved, they will be distorted. If you are remapping them back onto hemisphere in a QTVR-style presentation, rather than a 2D presentation, then there won't be any distortion; you're remapping a sphere onto a sphere. You just need to take shots from looking straight up to looking straight down, which can be pretty labor intensive.

In contrast with @Greg below, I like to take lots of photos in my panos and then reconstitute them all at once, on the presumption that more sampling will lead to smoother transitions between photos. That's only true if things aren't moving; in that case, the 8mm fisheye he recommends is an interesting choice, because then only a few shots are needed. Fewer shots = fewer chances for things to move around between shots.

To avoid problems with nadir check the following technique (BTW, the whole technical section worth examining also).

To avoid problems with color (maybe this will help you with zenith shots) make sure you either in manual mode (it will give you equal exposure in all shots) or at least aperture priority mode (it will guarantee you at leas same depth of field and exposure could be blended and equalised by the stitching software). In manual mode you should set desired aperture (it's good to calculate hyperfocal distance) and measure the brightest and darkest parts of the scene. This will give you the middle point of your exposure. But it seems you trying to shoot HDR, so this tip is not applicable in that case.

And one more. If you shoot the blank sky without clouds then stitching software could be unable to snap on the image while calculating the control points. At the current moment I can't say how to deal with such problems (it should be possible if we shoot using the panoramic head, but many software doesn't take this fact into considerations and tries to stitch by applying it's own logics). At least Kolor Autopano Pro/Giga and PTGui are do not consider the possibility of using panohead and base it's calculations on that fact.

Make sure you're tilting in pitch around the NPP point as well as rotating in yaw around it. You may have one arm calibrated correctly, but not the other on your panohead.

Having a separate piece of software, like Pano2VR, that can map your equirectangular to cube faces, so that you can then patch a specific face can also be helpful, if the head was over a featureless area--simple cloning/patching can erase the tripod.

If there are distinct features in the nadir, then you need to have additional nadir coverage aside from the shots on the tripod. Some people recommend moving the tripod, and then rotating the vertical arm of the head, so the camera is facing down for the nadir shot over the edge of the panohead (rather than the center of the tripod), but a handheld nadir shot (as long as the shutter speed is good) can work, too.

Don't include this "loose" nadir in the initial stitching in PTGui, but add it after the main alignment/optimization work has been done, and then use the viewpoint correction feature in PTGui to "patch" it into place. John Houghton has a good detailed tutorial on this method.