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.

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 $9.04808\cdot 10^{30}$.)

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.

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 $9.04808\cdot 10^{30}$.)

Update March 2023

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

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.

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.

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 $9.04808\cdot 10^{30}$.)