Gmail – Google Mail Fetcher not fetching emails

The e-mail fetcher would only fetch new e-mails that are on the inbox of the POP account.

IMAP is the the way to go. Here's a quote from another guide:

It was simple: I merely added Gmail as an IMAP account to my Apple Mail client, then just dragged folder-by-folder all the messages important to me onto my Gmail account. It took a while to transfer all the messages (be sure to open the Activity window in Mail to see your progress), but once it was done, all my old messages were safely stored in Gmail and had their original dates! Perfect!

So you want to be able to log in to your existing account, but be able to have emails to your company.com address show up there, and reply to those emails as if it's coming "from" your company.com address?

In that case, log in to your company.com account, go to Settings, and choose Forwarding/IMAP. Under Forwarding, choose Add a Forwarding Address and add your Gmail account.

Then, in your Gmail account, go to Settings, choose Accounts and Import, and under Send Mail As click "Add an email address you already own". Then enter your company.com email. Finally in the same section choose "Reply from the same address the message was sent to".

That said, if someone looks at the email header of mails sent this way, they will see it originated from your Gmail account and was sent "on behalf of" someone else. If that's not what you want, you'll have no choice but to switch over to the company.com account and use that "natively" instead. However in that case you can still import your old emails, filters, etc.