Question 1: The $L$-functon $L(E/\mathbf{Q},s)$ is a product over $L_p(E/\mathbf{Q},s) := 1/(1-a_pp^{-s}+p^{1-2s})$. Now plug in $s=1$ and use $p+1 - |E(\mathbf{F}_p)| = a_p$. (This is only a heuristic.)
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