Do not import your old wallet private key but to generate new seed phrase on your new cold wallet.
You can add passphrase which makes your coins to be more secure and safe. But do not lose the passphrase either, just as the seed phrase too. Both should be kept safe and securely in different locations. Also having two or three backups.
If you lose your seed phrase, you coin would be lost. If you lose your passphrase, your coin would be lost.
You're spot on. The safety of your coins depends on keeping both your seed phrase and passphrase secure and in different locations. If you lose either one, you won't be able to access your coins.
When generating a new seed phrase, it's advisable to use a trustworthy and secure wallet that has undergone rigorous testing and auditing. Also, be sure to verify the wallet address you're using to avoid being scammed by phishing.