Update, some recent information on (1):
Kalpokas, Korolev, Steuding recently released a preprint showing that $\zeta(1/2 + it)$ takes aribtrarily large positive and negative (real) values; and also show analog statements for the other lines through the origin, that is positive and negative (real) values of arbitary says of $e^{-i \phi} \zeta(1/2 + it)$ for any $\phi$. The paper contains also more quantitative results along these lines (cf. in particular Corollary 3 and the preceeding discussion).
Since (1) already received several answers, I expand and upgrade the comments on (2):
Yes, indeed it is conjectured, but unproved, that $\zeta(1/2 + i t)$ for $t \in \mathbb{R}$ is dense in the complex plane. [Side note: It is well-known that this is so for the lines $\sigma +it$ with $1/2 < \sigma < 1$.]
It seems that this conjecture was first formulated by Ramachandra (Durham, 1979), however only appeared in print in the second edition of Titchmarsh's book (note's by Heath-Brown), see the articles below for details.
There is very recent work on this problem due to Delbaen, Kowalski, and Nikeghbali. See in particular this preprint by the latter two and this by all three. Among others: in the former, they show how this result would follow "from a suitable version of the Keating--Snaith moment conjectures"; in the latter, they propose a refinement of the density conjecture, a quantitative version (see Conj. 1, in Sec. 3.9).
