I'm using Apple Mail, which was the source of all the trouble in the first place.
When you delete the account from Apple Mail and go to put it back in, *make sure* you go slowly and read everything on the screen. Make sure you are setting up an IMAP account (there are two buttons near the top of your dialog box - POP3 and IMAP - click the IMAP one). After you've set everything up, go back into the advanced tab of the mail accounts preferences and make sure everything is as it should be. For the USER, you will want to put the whole email address. Apple Mail just puts the part in front of your @. You will want to authenticate with your password, but not with SSL.
IMAP does not use SMTP. For HostGator, just put mail.[yourdomainname].com (unless it's .net or whatever) for BOTH the incoming and outgoing mail servers.
If you still have problems, I will create an IMAP email account (it's been a couple of months since I did this, and I'm a designer, not a back-end person) in my Hostgator control panel, add it to Apple Mail, and make note of everything I do and the settings, and I'll respond here. Try the above first, though. Use the list I posted of the commonly used ports. Sort out what you are going to do - IMAP or POP3, and use the appropriate port numbers. You have POP and IMAP settings all mixed up.
It doesn't make sense to set up a cron job for IMAP if you're not using IMAP.