Sandy,
Good afternoon, I wanted to address a couple questions you brought up, and I think the community might be interested in as well.
Q: If I run a site publish in the middle of the day at 1:00pm, and someone is working on a file and hits approve at 1:01pm will it push that file out?
A: How the publisher works now is on a queue, FIFO (first in first out). So the first publishing job will run, and complete. Then the queue will take the next persons job, and publish that one page on demand after.
Q: If I run a site publish, it says running, and then I close down my browser, does the publish still run?
A: Traditionally, this would not affect your publishing. Since Percussion runs service on a server, and it is outside your personal computer, you closing your browser should have no impact on publishing or the live first import.
With regards to Percussion not publishing certain items, and there being no clear cut errors on the publishing log screen shots, it leads me to think that the content might not have been submitted through the workflow, and/or a full site publish run.
The reason I suggest that is you mentioned you are able to publish on demand, that’s perfect. This proves the publishing mechanism is working. With misc images/css/content that doesn’t show up, I would assume you changed an image/asset/CSS or something misc. For this I would reccomend you go in and physically make sure that content is approved, and after that, run a full site publish. This should in turn publish out all the extra misc content that may have been changed.
If this doesn’t fix the issue, I am happy to take a look at your logs. Feel free to fill out a support ticket and we can dig deeper into the issue.
Have a great weekend 
Chris
