"I've found very little in the area of practical application of the Hypothesis..."
A simple check by Googling would confirm that (my attitude is that if it isn't on the internet, it doesn't exist). When you say "practical application", do you mean within math or outside of it?
"...and is evidently somehow related to modern financial security systems."
That would be because of the primes.
It would help to know something of your math background to try to answer your questions.
There really aren't many uses for the RH outside of math with the exception of cryptography perhaps. The reason is because cryptography utilizes the "randomness" of the prime numbers and proving the Riemann Hypothesis is equivalent to putting a really tight error term in the prime number theorem.