@ 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*:

\\ Pari/GP:
ri_zeta(t)=local(tmp);tmp=zeta(1/2+I*t);return([real(tmp),imag(tmp)])
ploth(x=0,100,ri_zeta(x),1)

(source)

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:

\\ Pari/GP:
ri_zeta(t)=local(tmp);tmp=10*zeta(1/2+I*t);return([tanh(real(tmp)),tanh(imag(tmp))])
ploth(x=0,100,ri_zeta(x),1)

(source)

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]

\\ Pari/GP:
ri_zeta(t)=local(tmp);tmp=1000*zeta(1/2+I*t);return([tanh(real(tmp)),tanh(imag(tmp))])

(source)

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.