First, are private keys for Non-custodial wallets to be looked out for and stored/saved just like the seed phrases? And
Private key and Seed phrase is just the same use they just vary on the format. As the word already implies, It’s the only key to access your wallet so you should save it or else your funds can’t be recovered or in event someone have access then your funds loss
Secondly, what other times would a wallet owner need to personally use the private keys aside from its primary automated role of authorize transactions?
You only need to use it to unlock/access your wallet then signing transactions is automated once you already imported the key to the wallet. You only need to import your keys once to access your wallet and send transactions. You only need to do that if you sign out or import on different non custodial wallet.