Picking zero out of zero elements

You simply did not set up the probability question right.

The probability of selecting ONE blue candy from five green candies is

$$\frac{0\choose 1}{5\choose 1} = \frac 05 = 0.$$

It isn't $\frac{0\choose 0}{5\choose 1}$ as you wrote.

To get a probability using ${0 \choose 0}$ you should calculate the probability of getting zero of something that doesn't exist: what is the probability of choosing zero blue candies from five green candies.

That'd be $\frac {0 \choose 0}{5 \choose 0}$ which would be $\frac 11 = 1$. You will always choose zero things if the things don't exist.

Colloquially: $n \choose 0$ is the way to choose zero from $n$ and there is always one way to do that: to not choose anything. To choose zero from zero is simply to not pick anything from an empty set. The one way to do that is to not pick anything.

First, how many ways are there to choose a subcommittee of $4$ Republicans, $3$ Democrats, and $1$ Libertarian out of a committee of $8$ Republicans, $6$ Democrats, and $2$ Libertarians? The answer is $$ \binom 8 4 \binom 6 3 \binom 2 1. $$ So how many ways are there to choose a subcommittee of $4$ Republicans, $3$ Democrats, and $0$ Libertarians out of a committee of $8$ Republicans, $6$ Democrats, and $0$ Libertarians? This should be $$ \binom 8 4 \binom 6 3 \binom 0 0. $$ But it should also be $$ \binom 8 4 \binom 6 3, $$ since you can just omit all mention of Libertarians in that case.

The number of $0$-element subsets of $0$-element set is $1$. Indeed, the only empty subset of $\emptyset$ is $\emptyset$. :-)

There is only one way to choose $0$ objects from any sized set - To not choose. That makes as much sense if the set itself is empty as if the set has multiple objects.