I can't settle for an alternative that's not vertically 360 degrees (or at least much more than 180 degrees).

A camera that shoots in every possible direction is said to have a field of view of 360 (horizontal) x 180 (vertical) degrees. Having more than that means you will be capturing some or all of the scene twice. Consider an imaginary arc that spans 180 degrees, from the top to the bottom. Now rotate this arc 360 degrees horizontally and you have covered the whole sphere.

What I would like to see: - decent resolution: at least 1920x1080. - and 30fps

For a 360 x 180 degree panorama the width will always be twice the height. Resolutions that make sense are 1920x960 or 2160x1080. You can't have exactly 1920x1080 unless you stretch, squeeze, crop or letterbox.

I've seen the ball camera, which isn't out yet, but would be really well suited for my case, given it does video as well.

If you refer to this product, then I don't see any mention that it records video. All it seems to do is take a single panorama when it reaches the highest altitude.

Another similar product is the Tamaggo 360-imager. It isn't out yet either, and while it captures 360 degrees horizontally the vertical range is less than 180 degrees. And it doesn't do video.

Yet another panoramic camera, the DIY Streetview Camera System, is a much better (and expensive) system than the other ones. Based on the example images on their site it shoots much better quality panoramas. It cannot do video, unfortunately. The fastest rate it can shoot at is one panorama every 3 seconds.

I think with time and patience one could build a DIY 360x180 video recording system. You would mount a bunch of small video cameras (maybe video enabled point & shoots or GoPros) to cover every possible direction with some amount of overlap. You want cameras that can be set to manual mode, since you want to get a consistent look from all of them. For syncing the cameras you could build some sort of controller that signals the cameras to start recording all at the same time, or else just start them manually and once all the cameras are running clap to get an audio cue that can help you sync the videos during post-processing.

To process the multiple video streams into a panoramic movie you first need to calibrate. For this you would take one set of images and build a panorama manually using, say, Hugin or Panorama Tools. Once you have a panorama project file that contains all the stitching parameters you can take the videos, break them into sequences of individual images using ffmpeg, run each set of images through the calibrated Panorama Tools project, and finally assemble a new movie with the stitched images using ffmpeg again.

Let me know if you make one of these ;-)