Ok, so I have dealt with my fair share of email calls during my ISP support call center days to know that technically it's possible to get close to what you're asking, but it can quickly get confusing as there are a few options, and important settings to be aware of.. hopefully I can give you a good overview to start.
There are two account types: POP3 and IMAP - this is for the account that manages email, not the email itself as it can be both.
POP3 downloads all the emails onto your computer drive. It doesn't handle multiple sessions very well and can cause connection issues for other IMAP devices sometimes, or often, depending on your luck.
IMAP is only able to read the emails that are located on the server, it never actually downloads a copy to any of your drives. This is perfect for smartphones for example, and IMAP handles multiple devices simultaneously much better.
By default, when you use a POP3 account, it downloads all the emails onto that computer, but does NOT leave a copy on the server, as to make sure the server never gets full, but when using outlook or any device and client for that matter, you can set it to "leave a copy of messages on server".
So what happens here, is it downloads a copy onto your POP3 computer, and still leaves a copy on the server for any other IMAP devices to see it.
With this basic understanding of how POP and IMAP inner workings, I think it's possible to get very close using the following setup. Anyone is welcome to correct me if I am wrong, but this is the best way that came to my mind.
Main computer set to POP3, leave a copy of messages on server turned on. BUT you can also set it to "Delete from server when deleted from this account." This means that if you delete it from this main computer, it will be gone from the server, and no longer available to any of your IMAP mobile devices.
Mobile devices and secondary PCs all set to IMAP, this way they only see what is on the server. However here is the tricky thing about it. If you delete an email on ANY of the mobile devices, just like the POP3 PC that is set to delete it from the server, it will be gone from the server for good.
The problem with this scenario, is that say you delete it from a mobile device before the PC got to connect to the POP3 server and download it, the PC will never see that email. So before deleting anything on the mobile devices, you need to make sure your PC already has downloaded it.
Otherwise, that email should still be in your trash of course, but no matter what device you delete it from, it will be gone from the server, and if you delete in from an IMAP account, it goes into your ONLINE HOSTED trash bin, not your local outlook POP3 bin.
I hope I have not confused you so far, but this should be enough to get pretty close.
You can also opt to not have the POP3 PC delete from server if deleted from the PC computer, this way, the mobile devices are the only ones whom can delete stuff off the actual email account - which is what I mean when I say 'online server'.
Finally, as I mentioned POP3 accounts can cause IMAP accounts to be unusable when they are connecting to download or send emails. This is usually temporary, and since you're the sole user of the email account, it removes the hassle of having a POP3 and IMAP user trying to use the email simultaneously.
However, sometimes outlook and other email clients can keep the POP3 connection hung, and even after closing it, it can take awhile before IMAP works properly again.. you might never encounter this issue though... the important part is to understand that IMAP = your actual online email account, POP3 downloads a copy to your computer drive, and that you don't delete from IMAP before it gets to the POP3.
Edited by dinotech-stjames, 04 March 2017 - 06:58 PM.