I'll answer your questions about Rabin-Miller, as well as put in a few comments about Diffie-Hellman. You have other questions in your comments; you can ask those in other questions (or just search through crypto.stackexchange for the answers - you aren't the first one to ask them)

As for Rabin-Miller, it has this property:

If $N$ is prime, then the result of the algorithm is always "probably prime", no matter what value of $A$ you selected

If $N$ is composite, then the result of the algorithm is "composite" for at least 75% (3/4) of the possible $A$ values within the range.

That is, if your $N$ is composite, and you selected $A$ *randomly*, then with probability at least 0.75, the algorithm will say "composite".

And, if you run the algorithm $k$ times, and select a random $A$ each time, then with probability at least $1 - 1/4^k$, the algorithm will say "composite" at least once.

These hold for any composite. Note that this is strictly true if you use random $A$ values; if you use a fixed value for $A$ (say, 2) the proofs do not apply.

That is what is meant by "running Rabin-Miller multiple times, you get a higher assuredness of primality".

That said, it is sometimes overly conservative. If you were handed a value $N$ from an adversary, and you are asked to test it for primality, and he "wins" if he gets you to accept a composite value, then this is the best we can do with Rabin-Miller; there exist composite numbers for which the probability of a false acceptance is very close to 1/4. On the other hand, such values are actually fairly rare - if you are testing 'random' values for primality, the vast majority of composite values will hardly ever come up with 'probability prime', and so it's safe in that scenario to use a far smaller number of iterations.

Now, on to Diffie-Hellman: it is often not safe to just use a random prime. The reason is that the security of Diffle-Hellman depends on the factorization of $N-1$; if you just pick a random prime $N$, $N-1$ may have a number of small factors (in addition to the factor 2 which is always present), and (depending on how you use it; how you select $G$ and the size of the private exponents you use), this can subvert security.

If you don't know what you're doing, you're better off using the RFC 3526 primes (which are known not to have any small factors other than 2)