1. I am pretty sure to update PI Graphics, PI Web Parts will sign up for updates with PI Update Manager for the tags used within the Graphic.
2. When referring to session time out, is there a specific error message you showing? If so, what is the exact wording of that error message.
3. Yes. For the PI Graphic Web Part, select "Edit Web Part" within the "Time Range" category check option "update every X secs" and supply a value for X.
If I understand correctly, there is no user interaction with the Web Parts page. It times out after 1~3 weeks.
Is the application pool the SharePoint / Web Parts running under recycling regularly? If I recall correctly the default is each day at 3:00 AM in the morning. What setting are you using?
I suggest to ask OSIsoft Technical Support to help you with investigation. If you like we can ask them getting in touch with you. Please let us know.
For the troubleshooting itself, I suggest using pilistupd.exe command line utility on the PI Server to check for the signups with PI Update Manager. I would also look at the PI Message Logs on the Web Parts server and the PI Server node around the time the timeout occurs. I am pretty sure there will be additional insights what the issue is caused by.