I saw these posts, and I just have to correct it. I work for a competitor of Hostgator, but I'd hate to see you switch hosts, deal with all of the frustration of a website transfer, then become disappointed when you find out it's actually nothing to do with the host you chose, even if it means we gain another customer.
First off: All email servers have a limit on IMAP connections. IMAP takes a lot more CPU power server-side than POP does. To make it worse, a lot of email software, mac mail being the biggest culprit, open way more connections then necessary, making it far worse.
1. Is Apple Mail behaving in a way that is not compliant with imap standards?
YES. Very much so. All email products made so far by Apple are abusive to IMAP servers. While Mac mail opens more connections than are necessary even when not malfunctioning, I've actually encountered some customers before whose Mac Mail clients decided to go nuts and open a jaw dropping 480 or so connections, before finally halting courier, the imap server software. (This was on VPS hosting, which had a very high limit) It's sort of ridiculous actually, since I've programmed IMAP software in Java which performed better.
2. Would this be a problem with other shared hosts?
Yes. No question about it.
3. Is there any way to control the number of imap connections made concurrently, in a way similar to thunderbird?
Not that I have found so far, sorry. We attempted limiting the connections by disabling the IDLE command with some customers, however this ended up not making a difference. I bet I could hack the software to make it happen, but I don't have a mac to tinker with. I use Linux only...
4. Any other way around this problem without switching to another email program?
Here's the good news: YES
If you reconfigure Mac Mail to use POP3, you can instruct Mac Mail to leave a copy of the messages on the server, so that Mac Mail behaves as if it was connecting using IMAP, minus a few insignificant features.
To configure POP3 to leave messages on the server in Mac Mail
1. Go to Mail -> Preferences
2. Click the accounts tab, then the account on the left, then the advanced sub-tab
3. UNcheck "Remove copy from server after retrieving a message"