I'm 13 and very interested in world building and creation, recently having an idea causing many questions seemingly unanswered by the internet. I am aware that a planet can run an unstable figure of 8 orbit between two stars. Let's say that this planet is inhabited by a humanoid race. Hypothetically, if one star died and produced a relatively small supernova:
What would the inhabitants see?
How would this affect their planet?
How would this affect the solar system and orbits as a whole?
There is no such thing as a "small" supernova, especially not for a "stellar death" scenario as you're imagining.
In order for a star to produce a supernova upon death, it must have a certain minimum mass - in particular, about 10 solar masses ($M_\odot$), where one solar mass is, not surprisingly, a mass standard equal to the mass of the Sun, or very roughly 2'000 000 000 Yg (yottagrams, for comparison, the Earth is around 6'000 Yg.).
Such a star will be rather short-lived - only about 10 million years or so, which is far too quick for any sort of life to ever evolve in such a system, unless it was introduced from elsewhere, e.g. someone settled on the planet from another star system.
However, regardless of this, the supernova cannot be made arbitrarily small because there is, as mentioned, a minimum mass for the star that will die in it, and thus the smallest supernova cannot be smaller than the supernova that this mass of star would produce. And even at this level, the supernova is devastating - about $10^{46}$ joules, of which essentially all is initially released as neutrino radiation; 1% of this is soaked up by the star and blows it apart. The resulting mechanical and radiative explosion at $10^{44}$ joules is always going to be enough to completely incinerate everything in the system, including all planets. In particular, suppose there were a planet as far out as 100 AU, or 15 000 Gm, about 100 times further away than the Earth is from the Sun. Assuming the energy radiates out in a spherical pattern, one can find the intensity of energy that will be deposited at this distance by
$$I = \frac{E}{4\pi r^2}$$
If you take $r$ to be the given distance (take $E = 10^{44}\ \mathrm{J}$ and $r = 1.5 \times 10^{13}\ \mathrm{m}$), you will find that the total energy pulse will be around 35 petajoules per square meter, or roughly one hydrogen-bomb explosion detonating on every square meter of planetary surface area at this distance. Granted, this energy will not arrive instantly, but over time; nonetheless it would be sufficient to flay an Earth-like planet at least down to the mantle, and for anything closer in (and it couldn't be too close if you imagine it to be habitable since the 10 solar mass star will be very bright, especially toward the end of its life), it may even be enough to entirely evaporate the planet altogether. As a result, no life forms of any kind will survive; the effects of the supernova will be complete and total annihilation, the death of the humanoid race in full as well as all other habitation within the system, if any is present.
Instead of a supernova, you may be more interested in checking out a stellar superflare of the type produced by very low-mass "flare stars" (red dwarfs), which is a rather explosive event but not of apocalyptic proportions. There are plenty of planets that have been found orbiting such stars, and one could ask about a binary system in which one of the stars is such a flare star, or at least capable of putting out a similar flare as apparently they have sometimes been observed to occur with more massive stars.
Regarding your question about figure-8 orbits, they are possible in theory but infinitely unstable in that an arbitrarily small "nudge" of the planet off its orbit by anything at all - e.g. the gravity of another planet in the system, or even just an irregularity in the stars' gravity owing to them not being perfectly homogeneous or symmetrical masses, will cause it to deviate from that orbit and into a different trajectory, perhaps ending up being consumed by the stars or ejected. (The instability is in a way the same as trying to balance a sharp pencil on its tip.) Thus such a system will not form naturally or persist were it formed artificially, and thus will not likely be observed to be in existence at all. For a stable orbit of a planet in a binary system, the binary must be either wide enough (the two stars sufficiently far apart) that the planet can orbit only one of them without getting cooked or pulled into the other, or else very close together (more like Tatooine from Star Wars, as you named in the question) such that the planet orbits in a huge circle or ellipse around both of them.
