The problem with the server might be caused by a
cache problem or corruption. When was the last time
you used a system and user cache cleaning utility?
Other than the periodic daily, weekly and monthly tasks, which I run now and then from the Terminal, never.
I can't import a Mail mailbox,
Any error message provided? A common mistake people
make when using the Import Mailbox function is trying
to choose the mailbox file or folder instead of the
location that contains the mbox file or folder.
Nope, no error message, and I am aware of that common mistake, having been bitten by it myself at some point in the past. Mail tells me the mailbox was imported and can be found in the Import mailbox, which does not appear anywhere that I can find it. I've heard this report from at least two other people on this forum in the last month.
Messages with an IMAP type account aren't downloaded
as with a POP type account.
Yes, I understand that, but nonetheless, there are people I've heard from who have messages that appear on the IMAP server that do not appear in the in box in Mail. I haven't personally experienced this, but seeing the opposite makes me think that it's not just something stupid they're doing.
Don't get me wrong, I like Mail, it's just that I'm seeing a lot of concerning reports lately, and am experiencing enough of those problems to be worried about it.