No, iCloud does not (and technically can not) 'sync' external POP3 email accounts.
If you want your email to be 'in sync' across multiple computers and devices, it just needs to be setup as IMAP (not POP3) on every device. You do not need iCloud to do this, and if you don't want to use an iCloud supplied email address then it couldn't help you with this anyway.
Contact your email provider and ask them for the IMAP email settings. Not all email providers offer IMAP, although most modern ones do. If they don't offer IMAP, then you are stuck. POP3 does not allow 'syncing' of email across multiple devices.