Does someone know if for the Riemann Zeta function the following has been proven:

1) Re(Zeta(s)) has no zeros for Re(s) >1

2) The minima for Re (Zeta(s)) in the strip 1/2<Re(s)<1 does not tend to minus infinity as Im(s) tends to infinity.

3) Has the zeros of Zeta on the critical line been plotted against zero number ({Im(s) | Zeta(s) = 0 } against the natural numbers) and a function fitted to the result?