Downloading Emails locally with Thunderbird

I am using Gmail (web interface) for sometime now, and have well over 20 labels and some thousand mails there archived to different labels in Gmail. Now I want to have a local copy of all my mails and following points are important in the context:

1. The Primary mail access mechanism would continue to be Gmail web for me. I just want a backup of my mail account locally.
2. Ideally the mails should download locally in folders named after Gmail labels (I know this is possible via IMAP but probably not by POP)
3. After all my mails are available locally, I will delete most of them in Gmail to free up space and because I want to archive them. The mails should continue to exist locally and should not be deleted when I delete when from Gmail web interface.
4. I would be syncing my gmail account locally let's say every month. So, the new mails that I have sent/received during this period should come over to my local mailbox in the folders named after Gmail labels.

I do understand that Gmail maintains a single copy of email having 2 different labels and such email would get duplicated locally in the 2 folders and I am okay with that.

Essentially you can see I just want to archive my mails from the Gmail server to a local backup and then sync (one way from Gmail to locally) new mails at regular intervals.

For some points above, POP seems to be the option while IMAP seems for the others. I am really confused and need help in deciding which of POP or IMAP would suit me best.

I have currently chosen Thunderbird to be my local email client but would not have a problem switching to Outlook or anything else as long as I get my desired archiving functionality.