Regarding Question 1, for their paper *The zeta function on the critical line: numerical evidence for moments and random matrix theory models*, Hiary and Odlyzko computed 5 billion zeros near the $10^{23}$rd zero.  The last had imaginary part approximately
$$
1.30664344087959822199974045053551×10^{22}
$$

See Table 2 of 
http://www.dtc.umn.edu/~odlyzko/doc/zeta.moments.pdf

This seems to be the current record.


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Update: In “*Alan Turing and the Riemann Zeta Function*” by Hejhal and Odlyzko, 
in the book Alan Turing - His Work and Impact, Elsevier 2013, they write
“It is now known that the RH is true for … some
hundreds of zeros near zero number $10^{32}$”  (This is $t$ near $9.04808\cdot 10^{30}$.)


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Update March 2023

In [New Computations of the Riemann Zeta Function on the Critical Line][1], Bober and Hiary set a new record, computing zero number $n=10^{36} + 42420637374017961984$ with $\gamma_n\approx 8.10292\cdot 10^{34}$.

[1]: https://arxiv.org/abs/1607.00709