Are ranks of Jacobians over number fields unbounded?

The answer to 1. is yes. Take $J_0(N)$ the Jacobian of the modular curve $X_0(N)$ over the rationals. Since all elliptic curves of conductor dividing $N$ are factors of $J_0(N)$ and there are infinitely many isogeny classes of elliptic curves over the rationals with positive rank. My guess, just like yours, is that 2. is open.

The answer to Question 1 is already contained in the following paper:

ANDRÉ NÉRON, Problèmes arithmétique et géométriques rattachés à la notion de rang d’une courbe algébrique dans un corps, Bulletin de la S. M. F., tome 80 (1952), p. 101-166. http://archive.numdam.org/ARCHIVE/BSMF/BSMF_1952__80_/BSMF_1952__80__101_0/BSMF_1952__80__101_0.pdf

The corollary to Theorem 7 (see page 155 of the article) says that for any given number field $K$, there are infinitely many curves $C$ of genus $g$ such that $\text{Jac}(C)(K)$ has rank at least $3g+5$. (With a bit more work, one can get $3g+6$. And with even more work, for $g=1$, the author gets up to rank at least $11$.)