For the minimum distance of an algebraic code, do you need to check the distance between all the code-words?

Answer 1

Yes. Minimum distance means the minimum of all distances between any two distinct code words.

Usually, the number of code words is very large, so it would not be practical to check them all individually. Even on a fast computer, this may not even be possible. So, to find the minimum distance of a large code, it is necessary to use mathematical techniques to prove a theorem that gives the minimum distance. The theorem would be something like "for all x,y being code words such that x not_equal_to y, distance(x,y) >= D". How you prove that depends on the particular code that you are working with.