Okay, I have been playing with this and sniffing packets as well.
Here is what is going on.
Turn off your auto-check feature on the iPhone during the test.
Send an email to your yahoo account, within 10-40 seconds, the top of your phone (while email is running) you will see the "spinner" start churning and stop. If you send a "asdfasdfasdfsdf" "test test test" email, for some reason, it will not DING you that you have a new email. But, it did check.
If you send a LONGER email or forward a longer email you have in your inbox to your yahoo account, it will indeed give you the instant notification.
I have been able to repeat this time and time again. When you get a new email, Yahoo is sending some type of notification via the cell network to tell your phone... HEY, YOU GOT MAIL and it goes and checks the yahoo email account.
I would like to use my domain name, not yahoo. Yahoo does offer paid service that allows you to use your own domain at $10 per year, granted that is one email address. However, they make no mention of it supporting IMAP, nor is there any mention of IMAP on the free accounts either, but we all know via the iPhone it works.
But, I don't know how that works with the iPhone account, because you can't change your email address to anything different, unless Yahoo! changes it when you sent through their servers for paying customers with their own domain.
Without the ability to use the AUTHENTICATE XYMPKI functionality in the standard "other" IMAP setup, then your SOL for bypassing yahoo. I would imagine, yahoo WILL offer the ability to support this for paying customers on your own domain.
UNTIL then, you can also setup a auto-forward on your mail server to send emails to your yahoo account, get the instant notification, then go in to your regular email account and read/respond to the message, using the free yahoo account almost like a "notification" service only. This takes longer because you have to check your email and download it, but at least your know you got a new email very quickly.
I have NOT tried the forwarding feature yet to see if it works. Those who say it won't work by forwarding from GMAIL, this is interesting as maybe they are detecting the forward via the mail headers and trying to keep us from using their service for the very purpose that I just explain. I will give it a try and see.