D-T fusion produces far more neutrons per joule outputed than fission, and they are very fast. This means that if you build a reactor out of any old scraps you have there is a very good chance that it will get transmuted into something radioactive, possibly problematically so. This does not mean that you cannot build a reactor that remains non radioactive, simply that you have to be very careful of the elements that you use to build it. Even if tokamok fusion does not prove to be the way forward, the work ITER and similar experiments have done is valuable research on reactor construction materials for any D-T design. On the plus side those same fast neutrons can transmute some problematic elements into less problematic ones.

Generally tritium is produced by bombarding lithium with neutrons. Originally it was thought that only lithium 6 was good for this, but then castle bravo happened and they discovered that lithium 7 works just as well if not better, which is how they got the yield so wrong. Hydrogen bombs typically produce their tritium from lithium as they detonate, which is how they avoid the half life problem.

I think the davy crockett was a standard boosted fission device. Fission devices typically have an air gap as part of the design, and there is no good reason not to fill that with fusion fuel. The yield can be adjusted by changing the amount of fusion fuel in this gap. Worth noting is that in this design the fusion reactions do not provide a significant boost to the energy, instead rapidly increasing the neutron population which makes much more fission occur. At the 10ton yield the weapon was acting as a pure fission bomb, with no boost at all. It's relation to fusion tech is that in a thermonuclear bomb you need a fission bomb to start the fusion, and there is no point making this first stage larger than it has to be. This pushed research towards looking for the smallest bomb possible. The 'tiny' yield is because it only just goes super critical, rather than any fusion occuring.