This is a complex structure and i believe can be achieved using when rules.
You can configure a data page which runs as soon as the portal/ page is launched.
1-Based on some property in the data page, you can put a when visibility condition on the left pane to show. Assignments can be displayed on centre pane just as shown in case manager. You can use that functionality.
2 & 3- You would need a tab group for this where the home tab is the worklist/assignment screen and other tab is a dynamic container. Any new assignment would open in a new tab automatically. You can however select on click events on assignment to change value of the property used in when to collapse the side pane.
4- You can flush the data page and/or property on home tab, re launch the page. This should do the trick.
I may not be very specific but you can create a setup on these lines to achieve your requirement.