What do we know about the structure of $J_{0}(N)$ over $\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}}^{\frac{{1}}{{p}^{n}}}])$? What is known about the structure of $J_{0}(N)$ over $\mathbb{Q}[\mu_{p^{\infty}}]$?
More generally, what do we know about $J_{0}(N)$ over 
$\mathbb{Q}[\mu_{p^{\infty}},k^{1/p^{n}}]$, where $k\in\mathbb{Z}$?
Does it have infinite rank? Does it have finite torsion?
I am especially interested in the case where $N$ is the conductor of an elliptic curve with additive reduction over $\mathbb{Q}$ at $p$. (This curve acquires, of course, a Néron model with semistable reduction st $p$ over $\mathbb{Q}[\mu_{p^{\infty}}]$.)
EDIT:
In view of the previous comments, I would like to ask whether the $rank({J}_{0}(N))$ is still finite over
$$K(k)\colon=\lim_{\stackrel{\rightarrow}{n}}(\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}}^{{\frac{{1}}{{{p}^{n}}}}}])$$
 A: Theorem 14.4 of Kato's paper in Asterisque 295 (2004), on page 236, says:

Let $A$ be an abelian variety over $\mathbf{Q}$ such that there is a surjective homomorphism $J_1(N) \to A$ for some integer $N$. Then for any $m \ge 1$, $\bigcup_n A(\mathbf{Q}(\mu_{m^n}))$ is a finitely-generated abelian group.

(I'll leave the case of Kummer extensions to others, but this answers the question in the title.)
A: The case of Kummer extensions is not yet completely understood, though partial results are obtained by Darmon-Tian in "Heegner points over towers of Kummer Extensions", Canad. J. Math. 62 (5) 2010, 1060 - 1080, following conjectures made in Dokchitser-Dokchitser "Computations in non-commutative Iwasawa theory," Proc. London Math. Soc. 94 (1) 2007, 211-272 (for instance). The results here are a bit tricky to summarize succinctly: There is a "root number dichotomy" which amounts to characterizing the forced vanishing of central values by the functional equation, and this must be considered a priori. The settings of forced vanishing typically correspond to Heegner-like growth of Mordell-Weil rank, as in the setting of anticyclotomic extensions, and this is described nicely in Darmon-Tian (who study a variation coming from varying parametrizations by Shimura curves). Anyhow, one predicts systematic growth of rank in these settings, and boundedness of rank in the others (i.e. when there is not forced vanishing coming from the functional equation). Also, results in this direction (including Kato + Rohrlich) are typically shown by combining some kind of analytic nonvanishing results for central values (such as Rohrlich's "On L-functions of elliptic curves in cyclotomic towers", Inventiones (75) 1984, 409 - 423) with an "Euler system" construction/argument (such as Kato's, as alluded to in David Loeffler's answer). 
