I've 3 email accounts on my BB. Two are gmails (one private, one business) and one is from my website (info@www) I'm not using gmail app on the BB, just messages. I have 2 problems:
My business email on gmail keeps sending mail I've sent to recipients back into my messages inbox, even though I've set a filter for it not to on my Internet Service account (identical settings to the private one, which behaves as it should).
The website email info@www doesn't push messages through to my BB at all.
Sorry I can't help with the other question, but perhaps if you posted a few more details (settings, etc.) someone else will come across the post who can.