Yes, you can search across Departments. Using the Global Search on the right pane of [Module] Home Page and select "All Department" option. Refer to the image below.
To create SLA & setup Escalation to supervisors, Goto Setup > SLA Settings > Service Level Agreement and follow the steps. For more, refer SLA and Escalations.
Yes, support agents can be associated to multiple Departments. Goto Setup >Admin Settings >Users >edit the Support Rep and associate departments to the Support rep.
No, it is not possible to restrict a particular Departments for selected Customer Portal users. Once a Department is made "Public", all Customer Portal users can access that particular Department.