Many times, I've corrected carryover amounts internally and not filed an amended return, if the tax didn't change, and the IRS has never inquired. Every year is a separate bucket with the IRS and they do not verify carryover items. They're just not that sophisticated, and they may never be that sophisticated.
NO. Each filing season, we are able to e-file the current return and the prior 2 years, so up until the shutdown date we are able to e-file 2020, 2019, and 2018 tax year returns. Then, after the IRS reopens for the next filing season, we will be able to e-file returns for tax years 2019, 2020, and 2021.
https://www.irs.gov/charities-non-profits/charitable-organizations/private-foundations is a starting point and should reference more authoritative sources.