This thread is appropriate for the Nuclear Engineering forum.
Nuclear waste is primarily the fission products produced as a by-product of the fission reaction. The fission products are contained in the spent fuel, which could be discharge to a permanet (or semi-permanent) repository like Yucca Mountain (NV), or it could be reprocessed, whereby the unused uranium and the Pu-239/240/241, which is produced by n-capture in U-238, is recycled. The fission products and irradiated fuel structural materials would then by calcined and vitrified into a ceramic waste, which would then be buried in a high level repository (like Yucca Mountain).
There are other waste streams such as irradiated corrosion products and contaminated clothing which must be disposed in low or medium level waste facilities.
That's an answer in a nutshell to question 1.
2. Why would we want to use a pebble shape for future fission reactors?
Greater surface area facilitating heat transfer from the fuel to the coolant. The carbide or cermet fuel ecapsulated in pyrolytic carbon and metal carbides is considered quite stable and strong at temperature. On-line refueling may be a factor.
3. What issues are there with decommissioning of plants?
Sending the irradiated structural components off-site to permanent disposal.
4. Why does fusion and fission only occur in particular elements?
Binding energy per nucleon favors fusion in isotopes of very light elements, and fission in certain isotopes of heavy elements like Th, U and Pu.
5. Why does an extra neutron increase the electrostatic replusion when we go from U235->U236?
This question is at the heart of fission. Why do some nuclides fission, and others do not? The extra neutron in U-236 causes an unstable nuclear configuration. It is conjectured that internal 'oscillations' form within the nucleus, which 'deforms' into two masses of charge, which then repel each other. Note that the two charge masses become a variety of fission products, i.e. each fission reaction produces two new nuclei, and there
Binding energy, fission and fusion -
http://hyperphysics.phy-astr.gsu.edu/hbase/nucene/nucbin.html
Fission fragments -
http://hyperphysics.phy-astr.gsu.edu/hbase/nucene/fisfrag.html#c1
Nuclear structure concepts -
http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/nucstructcon.html#c1
Radioactivity -
http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/radact.html#c1
Nuclear energy concepts -
http://hyperphysics.phy-astr.gsu.edu/hbase/nucene/nengcn.html#c1
6. In the new reactors that are planned, they stay single phase (ie He gas throughout or supercritical water throughout where does the energy come from if they aren't changing phase?
Well the He gas stays hot and under high pressure. Similarly, supercritical water is 'supercitical' which means that it exceeds the triple point at which liquid and vapor can coexist, and the supercritical fluid possesses properties similar to both liquid and vapor. See -
http://nuclear.inl.gov/gen4/scwr.shtml for the supercritical reactor concept. The higher temperatures mean greater thermal efficiency, but they also mean potential corrosion and structural integrity problems, i.e. safety margins are reduced.
Generation IV Nuclear Energy Systems -
http://nuclear.inl.gov/gen4/index.shtml