l functions - How many L-values determine a modular form?

I think the answer to your first question is "yes." Suppose $L(f,s) = sum_{m} a(m)m^{-s}$ and $L(g,s) = sum_{m} b(m) m^{-s}$, and that $L(f,n) = L(g,n)$ for $n geq n_0$, with $n_0$ large enough that the sums converge absolutely. Then pick an integer $M geq n_0$ and weights $C_M(n)$ so that $sum_{n geq M} C_M(n) m^{-n}$ is $1$ if $m=M$, and $0$ otherwise. One can surely come up with such weights without too much trouble. Then $a(M) = sum_{n geq M} C_M(n) L(f,n) = sum_{n geq M} C_M(n) L(g,n) = b(M)$. It's not too hard to see that if two modular forms eventually have the same Fourier coefficients, then they are the same.

edit: After some further thought, I'm having trouble justifying the existence of those weights. I found a different solution that I'm posting as a separate answer.