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Is it true that you don't need to enter a password to access voicemail? Are the voicemails downloaded to the iPhone's memory or do they stay on the network?
I don't need to enter a password to access it, no. Although I was required to set one. So, it clearly remembers the password. As far as where the messages are, I'm pretty sure it's on the network, but I don't know for sure. There's no file system browser. I would vote "on the network" because when I see the list, and I click a voicemail, it will only play if I have AT&T reception (aka not most of my apartment).
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Does the iPhone always keep a local copy of all messages and attachments from all your IMAP folders? Is it constantly synchronizing in the background? If so, this could be a problem for people with large multi gigabyte mailboxes on their email server.
From what I can tell, it downloads around 25 emails per account, and you can choose to download more if you want (in 25-email chunks) as you need to. This probably is a big hit on searching, but in terms of keeping the device relatively clean, it's effective. It seems to sync when you go into the mail app, so it doesn't lose a lot of cycles doing it in the background. And in the foreground, it only does 25 at a time.
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Is it possible to beam contacts or other items from one phone to another using Bluetooth?
Not as far as I can tell.
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When you create notes on the iPhone and then sync with a computer, are the notes synced with any application on the Mac or PC?
Good question. I just made some notes, but haven't sync'ed yet. I'll look when I get home.