Is zero a prime number?
No, 0 is not a prime number.
Indeed, zero is divisible by all integers! Therefore, it does not match the definition of a prime number, which is to be divisible only by 1 and by itself.
Parity of 0
Zero is of course an even number.
Number of digits of 0
0 is a single-digit number, because it is strictly less than 10; 0 is in fact itself a digit.
How to determine whether an integer is a prime number?
To determine the primality of a number, i.e., know whether it is a prime number, several algorithms can be used. The most naive technique is to test all divisors strictly smaller to the number of which we want to determine the primality. First, we can eliminate all even numbers greater than 2. Then, we can stop this check when we reach the square root of the number of which we want to determine the primality. Historically, the sieve of Eratosthenes (dating from the Greek mathematics) implements this technique in a relatively efficient manner.
More modern techniques include the sieve of Atkin, probabilistic algorithms, and the cyclotomic AKS test.