Hey guys. I know this is way after the fact but I had a problem where I was receiving emails as late as a day after they were sent, sometimes not at all. I typically received them with an hour though. also, when sending emails from my hotmail acct on my phone it would send the same emails twice. Either way, it was really annoying as I was used to receiving them as quickly as my text messages. Anyway, I tried all the suggestions on this thread with no luck. Then I realized that my wife's S3 was set up differently than mine, so I looked and realized i set mine up incorrectly.
Mine was incorrectly set up as a "windows live" account when I first set it up (which I normally associate with hotmail). However, it SHOULD Have been set up as a Microsoft Exchange ActiveSync account. So I deleted the email account, and re-did it as an exchange account and the emails come in as fast as texts. Nobody at Samsung, Verizon, or anywhere else thought to suggest this. Hope it helps save you guys the hours it took me to figure it out!