Yupee, I got mine, too. Directly from LMS : lms.ac.uk/CasselsFrohlich.pdf

That is the analytic side, on the algebraic it has to vary as well. The $\mu$-invariant changes. For instance the $\mu$-invariant for 11a1 at $p=5$ is 1, while for 11a2, the $X_1$-optimal curve, it is zero. This means that the Tate-Shafarevich group of 11a1 grows exponentially in size in the cyclotomic $\mathbb{Z}_5$-tower while the one of 11a2 stays the same.

Nono, the $p$-adic $L$-function must depend on the curve in the isogeny class. It is defined to interpolate values of the form $$ \frac{L(E,\chi,1)}{\Omega_E} = \frac{L(f,\chi,1)}{\Omega_E}$$ where $f$ is the modular form attached to the isogeny class of $E$ and $\Omega_E$ is the Néron period of $E$. So the $p$-adic $L$-function varies as the period varies.

Yes, Idoneal, by definition the $p$-adic $L$-function interpolates excatly these.