New iMac 2012 continued multiple questions

[Onamuji]

Onamuji wrote:
"Sent" mail, like most of your POP messages, will only reside on the computer that you used to send the messages from. That's how POP works. Messages are not saved on the server or synced with other devices. With the default settings, as soon as a message is received on or sent from your Mac, it is deleted from the server.

(You can mess with POP settings to save your messages in your Inbox even after they are downloaded to your Mac, but this is a kludge that seldom works well in the long run.)

If you want your entire mail archive on the new Mac and you're not "migrating" the entire old user account or then you'll have to export your old mail from your old Mac and import it on the new Mac.