In general, the built-in views of the Scheduler have specific configuration that distinguishes them from each other. For instance, the Agenda view is configured to display tasks for 7 days only.
Any custom requirements that differs from the built-in configuration of the view will require building a custom Scheduler view. Here is a demo that demonstrates how to produce a custom view, inheriting from the built-in ones:
With that being said, all of your requirements will require to build a custom Agenda view. Please find my comments to the specific questions below:
#1: The rendering of the Agenda view can be changed only with custom view.
#2: The current prev/next navigation moves the range with only 1 day. To change that behavior build a custom view/
#3: This will requite producing custom view.
I would suggest you examine the how-to demo that mentioned. For any additional changes, you will need to review the implementation of the Agenda view and override the desired parts.
Please note that custom implementations, like the discussed one, fall of the scope of the standard support service. If you need further assistance, please contact our Professional Services
Join us on our journey to create the world's most complete HTML 5 UI Framework - download Kendo UI