CypherGhost wrote:
Does that mean they can check my voicemail, which is not stored on the phone but stored on my carrier's servers?

I don't think so - this would involved accessing something that you were not carrying with you across the border.

If your phone is linked to your email account, can they read your email?

They could read any email that was downloaded onto your device.

Does that include just the email already downloaded or does it also include the email that will automatically be downloaded when they turn it on?

I don't think the later has ever been considered by the courts. I imagine that standard forensic practices would avoid connecting the device to a network after it was seized. However, I suspect that once the device was seized, you'd have a hard time excluded evidence that it automatically retrieved.

Can they cause new email to arrive, such as password reset notices from other services?

This is an interesting question - beyond my knowledge of the law - possibly beyond the laws knowledge.

Even if they were not, there's a lot of case law where police who discover evidence "accidentally" (such as finding drugs on you when you are being arrested for some non-drug offense).

Right - I think the "good faith" defense would apply here and the evidence would be admissible even though it technically wasn't covered by the search authorization.

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