Gmail shows a wrong number of unread inbox mails

If you are testing the POP account by sending emails from your Gmail account, those emails will skip your inbox. It appears that Gmail considers emails to yourself as a special case.

Took me a while to figure this out! Solved when I sent a test email from my old Hotmail account.

You do not need both. I suggest using mail forwarding because you will receive mail from your old account more quickly than using the POP3 method. Forwarded mail seems to be sent nearly instantly, while POP depends on the new account to poll from the old account. GMail varies the polling frequency based on how often the polled account gets mail. The delay could be as long as an hour. I don't think POP3 will ever be as fast as forwarding.

POP3 forwarding has one advantage: it logs into your old account and keeps it alive. This does not apply for GMail, but some email accounts are closed if inactive. My .edu university account is closed if it is not accessed for a certain period of time. In this case it may make sense to enable both: to get undelayed forwards and keep your account active.