Does the bot need my private key?

No. Your wallet approves one start transaction. The bot signs freezes with its own session key, which J Tools creates and seals on its server for this run only.

Can I take the freeze authority back during a run?

Yes. The authority sits in a multisig where your wallet alone can sign, so Take the authority back works even if the bot is down. The bot sees the change and ends the run.

What happens to frozen accounts when the run ends?

They stay frozen. The freeze authority comes back to your wallet, and you can thaw any account with Unfreeze Account.

Which accounts does the bot leave alone?

Your own wallet, the wallets on your whitelist, accounts owned by programs such as pool vaults (unless you turn that off), current holders in new holders only mode, and accounts outside your amount filters.

How much does it cost?

0.002 SOL for each wallet the bot freezes. It comes out of a deposit sized by your freeze limit, and whatever the bot does not spend returns to your wallet. You also pay, once, a small rent for the multisig account and the network fee of the start transaction.