I use the Google Launcher - I've never found anything in a 3rd party launcher that compelled me to switch. 98% of my use is with an app of some kind open, so the launcher and app drawer aren't really changing my experience hugely.
I just realized this is happening for my yahoo mail (using the yahoo native app), too. I just opened it after having gotten no notifications for new mail for several hours... There were 3 new mails in there all from at least a couple hours ago.
Same issue here. I will sometimes go into my email and there are unread ones there that I never got a notification for, or it will quickly refresh on its own and begin pulling in emails that are hours old. Happens on my gmail account and the two imap accounts I use with the gmail app.