Not easy for a "Newbie" like me :-)
There seems to be two problems: 1. not able to fetch the mails, and 2. not attaching the images.
1. Even if I deactivate all plugins, besides postie, it will not fetch any mails (only manually).
Sometimes the fetching works ones. I haven't digged into finding the exact reason but probably when I deactivate/activate, or set the fetching occurrence, of postie.
So the remaining root cause of the problem must be either the newly changed Linux environment to LiteSpeed V6.1, or the also newly changed php to 5.3.21 (if not WordPress itself, I use 3.5.1 and theme Atahualpa 3.7.11).
Now I have given up and signed up for a free account at SetCronJob, as a workaround to trigger get_mail.php, for the time being at least.
2. The problem by not attaching the images is verified being related to W3 Total Cache. Besides that the images are not attached due to missing
src the titles have been added with "Uncategorized".
When I add "/postie*" (without the quotes) in W3 Total Cache "Database Cache" in the field "Never cache the following pages:" then it works.