For 2, if it's a password that the user chose, the site should never email it, because the user likely reused that password across many sites, and someone who snoops on the user's email (say a housemate) can get the password to a ton of sites.

If it's a password generated by the site, then it's actually fine to email it. Although you likely don't want it too early in the email that it would show up in a phone notification or in a body summary in gmail.