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.