Как точно электронная почта продвигает работу в телефонах неежевики?

Does this mean that each time I switch from one mobile antenna to the next 
(say i'm driving down the road) my phone will send a new IMAP IDLE command
to my dovecot server saying "Hey, my contact address has changed, 
please use this one".

NO - IP-соединение

Ваш телефон обменивается данными с мачтами сотовой связи, используя стек протоколов, отличный от стека TCP / IP, который вы используете для создания "информационных" подключений к Интернету. (например, стек s7 ). Когда вы перемещаетесь от мачты соты к мачте, стек IP игнорирует изменения. Ваш телефон по-прежнему использует один и тот же IP-адрес (обычно) все время (если вы не путешествуете долгим путем, а не только по дороге к следующей мачте, где ваш оператор связи выделил отдельный географический блок IP-адресов).

НЕТ - Единый маршрут по умолчанию

Что происходит, когда вы используете и WIFI, и 3G? Ваш телефон issue two IMAP IDLE commands?

No, your phone will only be using one of them to send/receive data on the Internet (unless you have some magical load balancing phone, I've never seen one, but hey, we can't all have seen everything!). It will pick a preferred one and install a default route into it's routing table via that medium (typically wifi of course).

Finally, do all mobile phones (with a data plan) have unique IP addresses?

NO - There's a shortage "ya" know ;)

This is very rare indeed, in fact I have never seen it only heard of it. So I don't believe it still. Giving each mobile user a public IP would be far to wasteful, the same as giving every computer in your house a public IP. You likely have as many Internet capable mobiles in your house as you do PCs. With one wifi AP they can all share 1 IP. No-brainer for "typical" home use.

If not (say the telco NATs you) how can my imap server initiate a 
communication to my device?

Single Port/UPNP - You don't need "full" IP visibility

In order for your phone to communicate with your server (over IMAP) you will connect to TCP port 143 (or possibly 993 if you are wise) from a random outgoing port number above 1024 on your mobile. The NAT device at the carrier can create a NAT state table entriy in their GCN device mapping your phones private IP and source port to your servers public IP and port number. This will explain NAT further. Essentially though, you only need to connect to one port, which is quite simple for NAT. Another option such as that often used in home routers is UPnP forwarding but this is unlikely used at the carrier level.

To Wrap Up - IP Changes

So, when you are constantly on 3G, as you roam from cell to cell or mast to mast, your private IP between you and the carrier might change but your public IP talking to your IMAP server should stay the same so the IMAP connection should stay active. If you change public IP mapping, the worst that can happen is that the IMAP server doesn't recognize the IDLE prompt to keep the connection alive, as it doesn't recognize this as the already authenticated session, so assuming your IMAP username and password are saved to you phone, it should simply re-authenticate from your new public IP (phone and mail app dependent, but this is the behavior I find 9/10 times).

The same should be true if you connect to a wireless access point. If the IMAP server has a strop you will just re-authenticate from your new IP and carry on as normal.