If your email accounts are IMAP, not POP3, there's no "sync" - IMAP pushes email when it receives it. Syncing an IMAP account may get you the email half a second sooner, but it was about to be dent anyway, So just set all your email accounts as IMAP for receiving. (All email accounts are SMTP for sending,)
As far as the icon, mine labeled email has a white envelope, sealed, with a red @ on it. I guess each manufacturer uses a different icon for the program. If you go to combined inbox and press the sync icon on the bottom (second one, next to the "write" icon, it should sync them all. (Consistent with what I said about IMAP.) (Or maybe they're totally different programs, with just the combined inboxes in common, and you have no sync icon.)