The questioner has broached a most important topic that I've seen little written about. To wit: How keep a permanent collection of your emails regardless of the policies and configurations of email providers.
The advantage of the POP protocol is its simplicity and clarity of operation. Using the POP protocol, an email client like Outlook or Thunderbird simply downloads all emails received at the server in your name and then optionally deletes at the same time, deletes them later. or doesn't at all. E.g., an email account set up in Outlook to use POP downloads all your new mail on the server mail and stores them in a PST file on your local computer. Thunderbird handles POP much the same way. You can always count on having all your mail using POP.
Not Sent From Here:
The most common shortcoming of the POP protocol is that it's a 1-way trip. Your device's email client fetches email from the server and that's it. The mail you create and send to others is sent via an outbound SMTP server which does little more than forward your email to the rest of the world. It doesn't ever access your mail box. It is your email client that stores a copy of sent emails in the SENT folder of that device, if the client options were set to do so. Since the sending email client knows nothing of your other devices and most likely cannot connect to them - let alone their email clients, your SENT emails stay on the sending machine, unbeknownst to your other clients.
A newer protocol called IMAP (Internet Mail Access Protocol) attempts to allow multiple devices to see the same things no matter what device you are using. When a client connects to an email account via IMAP, the client and the server communicate both ways. Like Microsoft Exchange or Lotus Notes, IMAP synchronizes the emails and folders on the client with those and the server. So, if you sent 1 message from each of 3 devices, all 3 clients would see all 3 emails in their SENT folders. Likewise for deleted emails in TRASH or saved emails in an ARCHIVE folder.
Trouble in Paradise:
The disadvantage of IMAP is that the definitions of how it works varies considerably between servers and email clients. For example, email clients using IMAP do NOT usually download and keep ALL your emails. Instead, an IMAP server is meant to be the holder of your emails. Your IMAP email clients only synchronize a configured number of emails, not generally everything.
The rules are also muddy as to what your clients will see and how much (and for how long) the server will keep your mail. This is a showstopper for professionals where most emails must ALWAYS be kept somewhere and you know where that. Who knows when Google, Microsoft, Yahoo, GoDaddy, 1&1, Comcast, ATT, Verizon, et al, will change their retention rules. Such tech details are seldom delineated.
What is needed is a methodology that incorporates the best of both POP and IMAP. We already have such beasts in the form of Exchange Server and also the confusing swamp known as Lotus Notes. Both will mirror (i.e., synchronize or replicate) everything on the server with your local client and vice versa.
Alas, both servers are very expensive, complex, and not for the everyday user. Outlook, though, will work with such beasts. Microsoft is now pushing Outlook.com which is a multitenant Exchange Server for the masses. Internet service providers like GoDaddy and others are now acting as 3rd parties who offer Exchange mail boxes to plebeians.
Simple but imperfect workaround:
One way to get something close to the best of both worlds is to use both protocols if your server supports both. Using POP for your Outlook or Thunderbird account on your real computers and IMAP on your cutesy PDA's. Android and Apple have multiple email clients that will handle IMAP.
In this scenario, your PC's might fetch email using POP to download and archive all your emails from the server. You also set the client to NOT delete the server mail immediately so that your devices using IMAP can still see your mail for a while. What you do is set the PC clients to "leave mail on server for x days". I set that to 90 days. That way, my PC Outlook will eventually connect to and pull the new emails then instructs the server to delete any emails older then 90 days.
This workaround still doesn't handle folders you may have created on the server. Only IMAP will fetch and replicate folders both ways. By folders, I mean TRASH, SENT, ARCHIVE, et al. Unless you are a folder freak, you probably are most concerned about the SENT folder. That is, you want all machines to see what all machines have sent out - not just from the machine in front of you.
The simplest way to replicate the SENT folder is to Blind Copy (BCC) anything you send back to yourself. This is normally done manually by you at the time you send a new email. Some clients are getting smarter about this. The excellent K9 email client for Android has a configuration option to automatically add yourself to the BCC field on all emails you send. In so doing, anything you send gets received back to you as another recipient. Since all your devices will then always get the new mail, regardless of protocol, you have effectively replicated your outbound emails with all your devices.
The BCC trick is not perfect. All email that you BBC yourself with will end up in the inboxes of all your clients - not your local SENT folder. Outlook's advanced rules mechanism can help fix that too by checking all new emails in which your address is on both the SENDTO and BCC fields.
Hopefully, IMAP will evolve to allow all the features of both POP and IMAP so that all mail is downloaded to your client, including all folders - not just the more recent emails which is how vanilla IMAP setups work today.
Let's say you have 2 PC's (Wintel + Apple) and 2 PDA's (Android phone and tablet).
Set up PC's to fetch using POP with Outlook or Thunderbird, e.g.
a. Set protocol to POP
b. Set Delete Mail on Server for x days. (I use 90).
c. Look for a setting that will automatically BCC you on emails you send.
Set up Androids to fetch using IMAP.
Since PDA's, tablets, and smartphones often have memory issues,
IMAP is better as it limits what it keeps n the device.
Hope that helps a bit.