Supernovae do occur all the time, if you are thinking on cosmic time scales. The number I have typically seen in the Milky Way is that supernova occur once per century, roughly. Maybe a little bit more often, maybe a little bit less. Relative to a the lifetime of a typical star, that is a very short timescale indeed.
Now, the relevant number of stars is not actually 400 billion. There are two basic ways of producing a supernova. One is with a very massive star. Massive stars, meaning stars at least about 10 times as massive as the Sun, are quite rare, relative to stars like the Sun and more low mass stars. The other is when a white dwarf experiences some sort of mass transfer (the details of which are still not understood) from another star, exceeds its maximum mass, and explodes. While binary star systems are quite common, not all binary stars are close enough that sufficient mass will be transferred from one star to the other.
You are correct that stars are dying all of the time, much more frequently than supernova are observed. The key is that when stars like our Sun die, they do not produce supernova.
One other thing worth noting is that we do not necessarily see all supernova. This seems on, on the face of it, ridiculous. Supernovae are extremely bright! However, since very massive stars produce supernovae, and also have very short lifetimes, some supernovae may be hidden because they occur while still surrounded by the remnants of the molecular cloud from which they formed. With more and more advanced telescopes, the likelihood of completely missing a supernova in the Milky Way now is pretty low, but there have been supernovae in the Milky Way in the past 400 years since SN 1604 that were not visible to the naked eye as a result of being blocked out by dust. The best known example is Cassiopeia A, which from its supernova remnant is known to have occurred in the mid to late 1600s, but was not recognized at the time. Looking around some more, it appears that another young supernova remnant has recently been identified (Youngest Stellar Explosion in our Galaxy Discovered) that was not seen by the naked eye.
The ejecta of a supernova does indeed move at a fraction of the speed of light (somewhere around the 10% mark). However, it does not remain at this speed forever. As the supernova ejecta expands outwards, it creates a shell of material that is actually gathering up particles in the ambient medium (typical interstellar densities are around 1 particle per cubic-centimeter, much higher in molecular clouds).
After a few hundred years, the supernova remnant enters the Sedov phase in which the velocity of the ejecta moves at approximately
$$
v(t)=\beta\left(\frac{E_0}{n_0}\right)^{1/5}t^{-3/5}\,{\rm pc/s}
$$
After a few thousand years, the remnant's velocity slows down to approximately the speed of sound of the interstellar medium (a few km/s)--at this point we cannot distinguish the supernova remnant from the interstellar medium. The material that was part of the star is mixed in with the surrounding interstellar medium, thus seeding it with heavier elements.
As for first-generation stars, typically this means the metal-poor stars (where metal-poor typically means $[Fe/H]=\log_{10}(N_{Fe}/N_H)<-1$) that we call the population II stars, as opposed to the more metal-rich population I stars. Rarely does it mean the cosmologically-old population III stars (note that we have not actually observed these, so they're still hypothetical; James Webb Space Telescope might be able to catch the remnants of these) which have a metallicity of approximately zero (purely H & He).
Best Answer
There is a useful rule of thumb for estimating supernova-related numbers: However big you think supernovae are, they're bigger than that.
Disproving the rule, a quick search suggests that our most probable next nearby supernova would be about as bright as the full Moon.
However,