As far as you question 1 goes, the better way to think about why the connection between K-groups and L-functions exists is by viewing the K-groups as "almost" being etale cohomology groups (i.e., identical up to the 2-torsion part).

There is much more to your bullet point 3 than you indicate. Kolster's survey that you reference has lots of good information about this. For instance, see the Lichtenbaum conjecture on p.199, which is, to your third bullet point, what the strong BSD conjecture about the lead coefficient at s=1 of the L-function of an elliptic curve is to the weak version that only describes the order of vanishing. As for other arithmetic significance, you can easily define $K_2$ arithmetically. It is sometimes called the "tame kernel" (which you might google), defined as the kernel of a certain symbol.

Ah yes, thanks. I had also found a congruence for Fermat primes which seems to similarly follow from this Corollary. He doesn't give anything similar to the other congruences I mentioned, but now I will think about if the idea in his paper can lead to their proofs. (These would still be quite different from my own.)

Thank you Luis. I have already looked through both of those papers, as well as some other by Takashi Agoh, but none of them seems to have congruences like the ones in my question. (Corollary 1 in the second paper you mention gives a congruence for $q_p(2)$ when $p$ is a Mersenne prime, but it doesn't seem to me to be related to the one I gave).

Do you not believe that many other functions were considered? Even just by Euler? Look at the size of his collected works! Mathematicians consider many more things than turn out to be beautiful, or interesting, or useful, and those that prove their worth stick around for us to learn about them. I like to think that given enough time, each useful idea would be discovered by someone. I think a more interesting question is WHEN an idea will be discovered. It seems that many ideas "have their time", the almost simultaneous invention of calculus by Newton and Leibnitz being the archetypical example

Beyond this, most work has been on proving the Equivariant Tamagawa Number Conjecture for Dirichlet Motives, of which the Rubin-Stark conjecture is a consequence. Popescu and Greither discuss this. You would probably also like the Proceedings from the PCMI conference on special values of L-functions from the 2009 PCMI when the become available.

I don't know enough to go below and give a real "answer". But I don't know of anyone revisiting the case of K imaginary quadratic. The reason is that most people have been trying to prove the conjectures and their generalizations over arbitrary base fields. For instance, see the spectacular recent work by Popescu and Greither. Even if someone were to revisit the case of K imaginary quadratic to do things using special functions, I would imagine that they would attack at least the Rubin-Stark conjectures (Rubin K., A Stark Conjecture "over Z" for abelian L-functions with multiple zeros).

It sounds like he is trying to bound the partial sums from below by half the harmonic series since for each n, there exists a prime p with 1/2n < 1/p < 1/n. Of course the problem is that multiple n's can lead to the same p so this won't work. In any case, the "Proof's from the Book" proof of Bertrand's postulate given by Erdos is simple enough that I don't think the above proof, even if valid, would be nuking a mosquito.