This [1] is a good comment explaining why we will never know all the primes up to the largest known prime.
I am having a surprisingly hard time finding the answer to the second part of your question. This site [2] samples the first 50 million primes, and the OEIS tracks this sequence as A000040 [3] but they provide a very short list. No luck finding the longest computed A000040 sequence so far.
EDIT:
The research on the Golbach Conjecture summarized here [4] found all the primes up to 10^18, that is the most I am finding so far.