OmniFocus does not check the mail account by itself. It requires Apple's Mail client. (Someone might have made it work with a few other mail clients -- check the OmniFocus Extras forum.)
OmniFocus for iPhone doesn't do any mail processing. Of course, if you left the desktop app running and it received new messages, they'd get synced over to the iPhone.
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OmniFocus for Mac only syncs to-do items, not events, to iCal. This feature was primarily created for syncing OmniFocus data with PDAs that know how to sync with iCal. In terms of how things will integrate with a particular application like Sunbird, I recommend downloading OmniFocus 1.1 and trying it out. (There's a 2-week free demo period.)
http://www.omnigroup.com/application...ad/sneakypeek/
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Another possibility to consider, if you're getting into writing your own PHP webpage and stuff, would be to add in a few applescripts, so your data could go straight to OmniFocus for Mac, rather than being routed through Mail.