Create a mail server which encrypts received messages instantly with a public key and the stores them encrypted. This seems fairly easily doable with procmail.
Similar to previous.
This public key does not need to be published: it is only used by the mail server and user.
Such a server can be somewhat safely be hosted in the cloud, providing high availability. An attacker can intercept unencrypted traffic to the server (they could always do that), but seizing the server itself is useless because the server does not hold the private key.
We then need a protocol with which the user can fetch the encrypted messages and decrypt them locally with the private key. All searches of content need to happen on the secure local computer, which might be inconvenient for small devices.
If the server does not store arrival information, here is a sketch of a protocol to determine what's new: Associate each message with a unique key, then proceed bit by bit, with the client asking how many messages the server holds with keys matching a certain prefix. If the number agrees with what the client already has, then there is no need to ask further about that prefix.
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