The iTunes Icon on the task bar;
You need iTunes Preferences:
Edit>>Preferences>>Advanced tab.
Can you find that? There is a checkbox near the bottom: "Show iTunes icon in system tray".
The safe mode start: You need to press ctrl shift and then open iTunes, keep holding ctrl shift until iTunes opens.
Any program that you have that makes use of your iTunes library could potentially prevent iTunes closing as it has to be open in order for another program to access iTunes info.
Another thing that can cause problems is Bonjour. You can try stopping the bonjour service and then opening iTunes - then check if it closes.
The following article include instructions for stopping bonjour:
http://support.apple.com/kb/HT2250?viewlocale=en_US
If you want a brute force way of looking for other things interfering with iTunes, try MSConfig:
http://support.apple.com/kb/HT2292?viewlocale=en_US
There is a link to the XP version on the page if necessary.
This is a diagnostic method, not a solution.