I've never used wireshark, though I'd think netstat would be enough?
That spits out a list of anything currently using your network.

Do a netstat (open a terminal and type netstat) before Kmail loads (if
possible), when it first loads before it starts checking mail (if
possible), when it is actually checking your mail, then several times
following it finishing the mail check (including as son as it finishes
checking). When it fails to work, do several netstats. It may help to
paste everything to a text file to look through manually, these lists
can be long if you've got a lot running on your machine. Post the
kmail lines, and anything else you find suspicious. Perhaps something
is clogging your network when it happens, or kmail is spawning a bunch
of connections and isn't closing all of them.

Also keep in mind that TDE is relatively young and very small (two
devs, we really need some PR) and that the devs haven't had much of a
chance to make a lot of releases. We are working on cmake porting (so
we can use cmake rather than ./configure with a broken autotools) for
3.5.12, and it is a high priority for the devs to get the papercut
bugs fixed, hopefully also for 3.5.13 but definitely for 3.5.14. Looks
like you're a bit steamed, but there's only so much a small team can
do at once, especially considering that KDE e.V. dropped KDE3
entirely.

Also may be worth it to try a different distro that supports KDE3 or
TDE, I know there are builds for Debian Lenny and Squeeze, so it may
be worth it to install those with no GUI, add the TDE repos, and
install from there. This will take a long time (X is large, and IIRC
the TDE quickbuild server has limited throughput), but having no other
KDE version present will help make sure there aren't any conflicts,
and if you're not going to use GNOME or LXDE, this can save some
space.

-- 
Kris
"Piki"
Ark Linux Webmaster
Trinity Desktop Environment Packager