I have a question about 360 degree photography.
Is there a camera, or a technique, that allows us to take a 360ºx180º photo without showing our hands or the tripod below the camera?
Actually, most of the 360x180 panos you see are created by taking multiple images and stitching them together as panoramas. (See: How are virtual tour photos taken?). Erasing the tripod is a combination of shooting and post-processing techniques.
Most of the tripod and panohead can be eliminated from the pano by simply shooting two nadir (straight down) shots, taken 180º apart in rotation. This allows you to have enough "clean plate" to use masks and layers to erase most of the panohead's vertical arm so that the tripod is in a relatively small circular area of the cube face.
Then, after you've shot everything, you take the camera off the tripod, or move the tripod and tilt it, to get a "clean plate" area where the tripod was, and then use that to patch over the area in the panorama.
I use the viewpoint correction tool in PTGui and the "clean plate" shot to cover where the tripod went, as I don't have to leave the stitcher and go into Photoshop to "fix" the nadir. The viewpoint correction can take into account the moving of the camera's viewpoint and the tilt downwards to cover the floor.
There are a variety of other techniques, without PTGui, to do the final erasure of the tripod or the blank area left from masking, such as mapping out the cube faces with a tool like Pano2VR or Hugin, or adjusting the pitch -90º to put the "hole-in-the-floor" in the center of the pano where there's the least distortion, and then using Photoshop/Gimp cloning or patch or content-aware fill or masks/layers to erase the tripod. If you set the tripod down on a relatively featureless "floor", you may not even need the clean plate shot, and can just use the patch tool to erase the tripod.
Some folks don't bother with tripod erasure. They simply cover up the tripod area with a round logo image. :)
Hands are typically not an issue, because you make sure you don't have your hands in front of the lens when you do this. Hats and shoes are a different issue, however, if you are working with a circular fisheye lens with 180º HFoV, but you can try to remember to take off your hat, and the "clean plate" thing works for your feet if you're hand-holding.
Using an 18-55 kit lens with an APS-C entry-level body is... not optimal for shooting these types of panoramas, and certainly not handheld without a panohead. There may be parallax issues, but coverage is the main problem. At 18mm, you will probably have to take far more images to completely cover the sphere for a pano than you anticipate.
The FoV of an 18mm lens on an APS-C camera in portrait orientation is roughly 45ºx63º. But you also need sufficient overlap to ensure a good stitch. To cover 360º in yaw, with the camera in portrait orientation (to get more vertical coverage) you need 10 images, and three or four rows, so 30-40 images minimum to cover the sphere, and you might have to add zenith and nadir shots.
Even a rectilinear ultrawide lens, like the 10-18 will still require multiple rows, as well as a zenith (straight up) and nadir (straight down) shot. Fisheyes tend to be the lens of choice here, and can simplify things down to 3-8 shots, depending on how precise you can be about rotating the camera around a specific spot in space, and how much overlap you need.
I got sucked into buying my first dSLR to learn how to do VR panos, because I realized a fisheye lens was much more convenient. It has a much bigger FoV than a rectilinear lens of the same focal length. If you're ok with manual lenses (no electronic communication with the camera, so manual focus/aperture and you can only shoot in M or Av and there's no lens EXIF) and don't plan to go full frame, the lowest-cost one you can probably find at the moment is the Samyang 8mm f/3.5. Also known as a Vivitar 7mm, Opteka 6.5, or 8mm Rokinon, Bower, Phoenix, Pro Optic, etc. etc. You can cover the sphere in 8 shots (6 around, zenith and nadir) with one. This is a different design and a completely different lens than the 8mm/2.8 (Sony e-mount/Fuji X) and 7.5mm fisheye (micro four-thirds) Samyang makes for mirrorless mounts.
By "360 photography", I'm making the assumption that you want a spherical view (aka "photosphere").
The Panono 360 (https://www.panono.com) is a throwable camera taking a complete spherical image, suspended in the air. With this kind of "camera", no hand/foot and no tripod in view...
If you want something more classic, you can use a camera on a tripod for all your shoots and remove the tripod for the one looking down (manually holding the camera). It shouldn't be very tricky to align afterwards.
You might also try using no tripod. Unless you are making Gigapixels panorama, 360 views don't require hundreds of images. You may be surprised by the quality of hand-holded panorama, given a little practice.
If the ground allows it, you can also try the "delete and replace" feature of Photoshop (or other tool) to erase the image of the tripod.
Regarding hands, the technique don't put them in front of the camera might help. Of course being invisible could also be helpfull (same thing goes for your feet).
The area directly beneath the camera in a spherical panorama is called "the nadir", and as you've pointed out, it will normally show parts of tripod and/or a black polygon for the un-imaged area:
Usually you either restrict the viewing angle so that the user can never see the nadir, or you "patch" it to replace the area with something else. Here is the same nadir patched with a company logo:
I wrote a free, open source panorama editor recently that's made specifically for equirectangular (360° x 180°) panoramas, and one of its functions is to patch the nadir/zenith areas:
Give it a try, could be the simplest solution to your problem.