When I power off the iPod, and restart it, does it re-open all previously open applications? On the last week I powered off the iPod at least once per day, but today I noticed that in the multitasking tray there were applications I have not used in the past weeks. Are those applications running, or the tray simply list them because I used them?

If those applications are really running, is there a way to stop the iPod from running the same applications that were running before I powered it off?
These applications are not necessary "running".  The multi-tasking tray doubles as a "recently used" list.   So although you may not have run these applications in the past week, they are listed in the order they last ran (however long ago that was).

So, to answer your question, no the iPod doesn't restart all previously open applications.