You provide the self-certification to the IRA custodian (your broker?). It's what allows them to accept a rollover after the 60-day rollover deadline has passed. It is not to be provided to the IRS unless the IRS asks for a copy.
"Lastly...my understanding is I should pay the early withdrawal penalty upfront, in order to get a refund of that penalty. Is that correct? That is, I need to "rollover the entire amount" in TurboTax."
If in TurboTax you indicate that the entire distribution was rolled over (which you will do if you complete the rollover of the entire distribution under Rev Proc 2020-46), there will be no early-distribution penalty to pay up front and no penalty to refund.
