Can we find the minimal crossing number for a given knot or prove that a given diagram or code for a knot is the one with the minimal number of crossings?
For the 8-knot it always has at least four crossings. Can that be generalized for any knot or any diagram or knot code, that the diagram of this particular knot implies that the knot has a minimal number of crossings?
Knot tables traditionally give (prime) knots according to their minimal crossing number. The way this works is that it's relatively easy to enumerate all knot diagrams for a given crossing number, so supposing you have a table for all knot types up to $n$ crossings, you can add in the knot types for $n+1$ crossings by enumerating the diagrams and throwing out all the ones that are equivalent to one in the table so far (this can take a fair amount of work).
Sometimes you can tell immediately whether a knot diagram has minimal crossing number. For example, diagrams of alternating knots (or more generally adequate knots) that are reduced have minimal crossing number (this is the first Tait conjecture, proved in 1997 using the Jones polynomial by Kauffman, Murasugi, and Thistlethwaite). Reduced means there are no crossings along which you can twist half the diagram to remove -- or in other words, there is no crossing such that there are fewer than four distinct regions of the plane around it.
Your example of the figure-eight knot is an alternating knot, and any four-crossing diagram of it you might have in mind is reduced, so four is the minimal number of crossings in any diagram of the knot.
As it happens, knot tables are organized such that for a given minimal crossing number, the prime knots are enumerated in the following order:
For eight-crossing knots, $8_1$ through $8_{18}$ are alternating knots (none torus knots), $8_{19}$ is a non-alternating torus knot, and $8_{20}$ and $8_{21}$ are non-alternating knots. This is using the Alexander-Rolfsen-Briggs notation -- as far as I know the subscript has no meaning other than the index of the knot in the table.
By the way, this is a useful resource for knot data: https://knotinfo.math.indiana.edu/