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.