Still hanging on to the phone (instead of sending it for service), trying to figure out what goes on. (Sorry to bother you with this, but I've figured out a little bit more that might shed some light on the issue for those of you that understand much more about this than I do.)
The problem e-mails are actually received by my phone as long as Outlook is not running on my computer. As soon as I start Outlook on my PC, the problem e-mails disappear from the phone. The problem e-mails seem to be (mainly?) advertisement/product information e-mails from non-personal senders such as AVIS (the car hire company), The Mathworks (the mathematics application company), Farnell (online electronics components company) and so on, with lots of URLs and pictures in it. In some of these e-mails, Outlook (on the PC) will prevent downloading pictures in the e-mail - in others Outlook won't (because I've added the senders on my "safe senders list").
Further, if I (after the e-mail has disappeared from my phone) change the read/unread status of the problem e-mail back and forth on my PC (and sync it with the Exchange server) and then run a manual "send and receive" on my phone, I can see (in the status bar below the list of e-mail messages on the phone), that the phone actually processes the e-mail, but it won't display it. (If I "send and receive" again without changing the read/unread status, no such processing can be seen.)
So it appears, after all, that Outlook on the PC is involved in the problem (all rules are still off and I've even disabled the McAfee anti-spam) and that some processing (in Outlook) of the e-mail occurs, which is then synced back to the Exchange server, and which in turn prevents displaying the e-mail on the phone after the phone has again synced with the Exchange server.
However, there is also something fishy in the phone, since the disappearing of the e-mail does not occur on the HTC (with WM 6.1).
Based on this information, any new ideas, anyone?