Going at it back to front...

2) Why do you want to do this? Your answer may well influence the answer to (1).

In Thunderbird, you can drag and drop messages, and maybe folders too, between accounts. (But I have heard of bad things happening when folders are moved. In some cases, it seems they retained their allegiance to the original account they were dragged from.) I wonder what benefit you seek from having messages moved or copied into different folders? Since both your accounts are POP-based, everything in Thunderbird is local and doesn't affect what's on the mail servers, so I am puzzled by your use of a separate email account as "a place to store copies of messages that I receive in my tbird application". If the messages are only viewed in Thunderbird, what's the point?

IMAP makes most sense if you want to access an account from multiple computers or devices or via a web browser. Your use of POP makes this appear an unlikely requirement for you. But if you want to build a complex folder system stored externally, useful as backup against local hardware failures, or because you want to access it from more than one device, then IMAP is the only way of seeing those folders on the server. POP shows you only the account's Inbox. It can't see other folders on the mail server.

Slightly confused - you mention you have two accounts: a pop gmail and a pop tbird however, what do you mean by pop tbird?

Thunderbird email client is not a server and does not issue email addresses. So you must have another email address on a different server.

Pop mail accounts: You can have both pop mail accounts incoming to their own mail accounts and using their own Inbox.

you can also set up Pop mail accounts to use a Global Inbox in Local folders account, so they use only one Inbox in thunderbird.

All emails can be stored on computer in Thunderbird profile folders. You can select to leave on server or remove upon download or keep on server for a specific period or remove when you delete an email.

Pop mail accounts are useful if you only operate from the one computer. As emails are stored in Profile you can view them in online or offline mode.

If moving about using different computers or devices, you can access mail via Webmail or need to make sure that Pop mail accounts leave messages on the server in order to download to separate devices. Sent emails cannot be seen by differnent devices as they are only on the computer that sent them.

You could use a portable version of Thunderbird to overcome this issue, but you may find an issue with Profile size if there is a lot of mails.

IMAP mail acounts see a remote view of the server. You subscribe to see folders and email headers are downloaded. when you select an email, Thunderbird downloads to a temp cache to facilate quicker access, but emails are not stored in Thunderbird, they are only stored on server. You cannot dowload to see emails if in offline view.

You can synchronise subscribed folders to get a copy of the server folder in a mbox file in your Thunderbird mail account. Very useful for offline viewing. Anything you do in an Imap account will be reflected in the server folders. So if you delete an email in Thunderbird or via webmail, the folders are synchronised to show same. They are one and the same folders.

As with Pop, you can move emails into Local Folders, these emails will be stored in mbox files in your Profile and are not efected by the server. If needed, these emails can be deleted off the server and still have a copy in Local Folders.

IMAP mail accounts cannot use a Global Folder. IMAP mail accounts are good if you use different computers or use eg: a phone and want access to the same information.

Whatever method you use, both email addresses set up to use their own mail account can be seen symultaneously in the same Profile in Thunderbird, so you do not have to forward emails to other accounts, that is your choice.

Backups are still inportant regardless of Pop or IMAP.