Do mining pools expose attempted nonce to the public ?
No, that's pointless and would be completely infeasible. The combined network tries over 100 billion billion nonces per second (as of december 2020, it's around 120 exahash/s).
I would think no because it would be counter competitive. I can let a giant mining farm tell me what nonces not to use
That's irrelevant. They are trying to find nonces for a different block candidate than you are (at the very least, they're trying to mine a block that pays them; you're trying a block that pays you).
Would it be faster to continue attempting random nonces
No, every block candidate + nonce is a completely independent attempt. Nobody will ever try the same combination again (because the timestamps change, the composition of transactions changes, and the payout depends on the miner). It doesn't matter in what order you or anyone else tries nonces - only how many you try per second matters.
or storing the nonces previously attempted in memory and finding them in a sorted array
No, same reason.