When we are doing the Statement of Cash flow, under which account in the T-accounts would we put a gain on sale of investments (Held-to-Maturity)?

A profit on sale of investments belongs in an other Income account of that name

How do we know if the accounting errors reverse themselves out or not?

You need to be more specific, what accounting errors? Errors found by auditors? Errors found by reconciliation? My experience is that errors are found by diligent inspection of the accounts and reports and errors are corrected by accountants