@ Q1: After the counterexample of Noam Elkies I used Pari/GP to draw that parametric plot to get more visual impression;
[update] The visual impression in the 1:1000 zoomed picture had artifacts; I deleted the picture and provide a more precise one and corrected in my original answer [/update]
Plot 1 shows the known curve in the complex plane, when t increases from 0 to 100:
From the drawing one cannot discern, whether there is some crossing of the negative real axis. Here is a rescaling; the values of the zeta-function are scaled by the tanh-function:
and then a strong scaling factor of 1:1000 applied. [update] To remove artifacts, there is an option "recursive" in the plot-routine to scatter the coordinates more regularly; the strong zoom separated the dots of the plot too much so that artifacts are likely to occur. With an improvement of the sampling no crossings of the negative real axis can be seen [/update]
I used internal precision of 200 dec digits, [update] so I think the computation of the single points do not introduce artefacts, but the connection by lines may do due to the strong scaling required. This type of plotting seems to require much resources; I'll see whether it can verify the crossing in the near of t=282 visually; I'll update then this answer again.