In answer to your first question, the fuel in standard nuclear reactors is Uranium-235. The chain reaction that produces the heat is started by bombarding slow moving neutrons at U-235 nuclei - http://hyperphysics.phy-astr.gsu.edu/hbase/nucene/fission.html
The speed of these neutrons, and the choice of U-235 is very specific to the reaction. The spent fuel roads in a reactor have used most of the uranium content, leaving more stable fission products that can't be induced into a chain reaction - http://hyperphysics.phy-astr.gsu.edu/hbase/nucene/u235chn.html
The fuel rods coming out of the reactor may be hot because they've come out of a reactor with a temperature of a few hundred degrees, but theyre no longer contributing to maintaining that heat since they're not a part of the chain reaction anymore. They'll quickly lose this heat once they leave the reactor, so theyre not much use in terms of energy production.
Fuel rods coming out of a reactor can be very radioactive, as the fission products are usually unstable. These fuel rods are often called 'hot' by nuclear physicists, but that's in reference to the energy of the radiation released from them, not their temperature, and this radiation is hard to harness efficiently into energy.
In answer to your second question, the heat produced in the reactor is a measure of the increase in kinetic energy of the particles within the reactor, and the transfer of momentum from these particles to the coolant passing through it. So essentially the process is already directly harnessing the momentum of the particles.