this isn't really a phone issue at all, but I'll be happy to help. you don't need to use imap to do as you desire, tho it may be the best solution. Verizon may not even support imap. most likely thunderbird is set to delete mail from the server after it's downloaded. I haven't used tb in many years so I'm very rusty. there should be a setting called 'leave mail on server' or something like that under your account's settings in thunderbird. I know the comparable setting in Outlook makes it delete and mark as read all email that's downloaded by default. if my hunch is correct, thunderbird is downloading your inbox, making all emails as read, and since phone email apps normally only display unread mail in your inbox, these read mails won't be present.
ideally imap is what you want to use as it will update your tb in real time and ensure tb and your phone have exactly the same emails. pop3 syncs with the server every x minutes (phone apps will use push to get emails right away) and stores copies of your mailbox on your computer, so it's possible tb and your phone may have different contents depending on how your various email programs and apps are configured.
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