IMAP and Mismatched Mail Servers

[#1]

I have a number of non-Plusnet email accounts accumulated over the years.  At the moment I use POP3 to receive directly from the appropriate server but send via relay.plus.net.
I have always been a bit hazy about IMAP and the way it works.  It's basically an incoming  mail protocol but, if asked, will also synchronize your sent items with the mail server.  I've assumed this requires some sort of co-operation with the outgoing mail server and therefore would only work if the two servers belong to the same mail provider.  So if this were true it would mean I could use IMAP for my Plusnet emails but not for my other email accounts because the incoming and outgoing mail servers don't form a 'matched pair'.  But I have never known if this is really true or just a misapprehension on my part.
Can anyone enlighten me? 

[#2]

Well, although your mail message goes through the SMTP server (whatever server that may be), once the SMTP sends a confirmation that email has been sent without error... It's actually uploaded to the sent folder using the IMAP server rather than through SMTP.
Edit to fix grammar issues. Can't seem to get it to sound right.

[#3]

A longer way of saying basically the same thing, and more.
I think IMAP is basically a mail access protocol, it doesn't do any synchronisation. That is down to the email client The same is true of POP3, it's up to the email client what happens to mail on the server after retrieval. One difference though, POP3 is one way (download), IMAP is two way - but only to and from the incoming mail server, one can't 'send' using IMAP.
For sending one uses SMTP to a relay server and mail clients are usually told to store a copy in a particular place. That copy is put there by a 'copy' process, and taken from the Outbox, strictly it isn't a direct copy of what went to the SMTP server.
Thunderbird can be set to store a copy of sent mail in the IMAP sent folder, and uploads the message twice to do that (once to 'relay' and once to the 'sent' folder. Outlook only stores the copy locally but one can write message rules that move messages arriving in the local 'sent' folder to another folder - and that folder can be an IMAP one - effectively achieving the same as Thunderbird.
With Thunderbird one can move messages in a non-Plusnet POP3 folder to a Plusnet IMAP folder, and (for example) forward them using webmail. I've done that on occasion.