So I just asked scottro (forum mod) to let me setup a domain on his hostgator account. I installed OSTicket 1.6ST and configured it minimally. (after initial install questions went to Admin panel -> Settings and enabled everything under "Autoresponders (Global Setting)" and "Alerts and Notices".
Then I opened a ticket. I got the open ticket message.
Then it occurred to me that maybe you meant you setup POP/IMAP on your implementation and cannot seem to get it to work. I've never tried to put my mail through hostgator so this will be ... long. After digging thru HG's support site I found:
http://support.hostgator.com/articles/hosting-guide/lets-get-started/dns-name-servers/how-to-point-my-domain-to-hostgator-when-using-dns-elsewhere(http://support.hostgator.com/articles/hosting-guide/lets-get-started/dns-name-servers/how-to-point-my-domain-to-hostgator-when-using-dns-elsewhere)
Now I'm waiting for dns to update to be able to test. While we wait I went into OSTicket to do some quick configuring. I went to Admin Panel -> Settings and enabled "Enable Auto Cron:" and then went to Admin panel -> email and added the new email address I setup. I also added the smtp server while here. HG's support document for client settings is located at:
http://support.hostgator.com/articles/specialized-help/email/outlook-eudora-thunderbird-mac-ipad-phone-client-settings(http://support.hostgator.com/articles/specialized-help/email/outlook-eudora-thunderbird-mac-ipad-phone-client-settings)
So to be a wise ass I sent an email from my new email account to itself (which of course generated a bounce message). Which resulted in OSTicket grabbing the bounce message and opening a ticket. So DNS hasn't updated yet that's fine.
After waiting awhile (honestly about 10 minutes), I sent a test email from myself to myself and OSTicket opened a ticket for it. I went to the /scp and responded to the ticket, and my test email account received the reply. So I went over to gmail and sent an email to my new account. Then popped back into /SCP and clicked around until autocron triggered.
If you have configured everything correctly (including your domain's MX records) you should have no problem utilizing hostgator as your email and web host.
PS: it is important to note that if you open a ticket with say ostest@gmail.com and that is the same address your using in OST that you never receive a message stating "A request for support has been created and assigned ticket #BLAH". Use a email that's NOT configured in OST to try to open a ticket.