Representing integers that are larger than the machine integer size by an array of digits is the standard way to do it, and it's usually called bignum, or more formally arbitrary-precision arithmetic.
Division on bignums is not simple. There's no shortcut: it isn't like with addition and subtraction where you can make a single loop that goes over both arrays. You can use the long division method you learned in primary school. There are more complex methods that have better performance in general; the Wikipedia article has a good overview.
There is no useful relationship between $n \bmod (a+b)$ and $n \bmod a$ and $n \bmod b$. I don't see where this would come up anyway. (On the other hand, $(a+b) \bmod p$ is either $(a \bmod p) + (b \bmod p)$ or $(a \bmod p) + (b \bmod p) - p$ depending on whether the sum overflows $p$).
When computing with numbers modulo $n$, always reduce modulo $n$ after each operation, to avoid having numbers grow too large. During one operation, it's usually useful to allow one more digit, to accumulate carries on partial results.
Note that primality testing is rarely done on large numbers, because it gets extremely slow. In cryptography, approximate algorithms are used, that have a small probability of accepting a composite number as prime (but if the algorithm says “prime”, then the number may not be prime, but it's “difficult” to factor).