User has applied project assignments in CORE to control the employees assigned to a project. This, in turn, controls the projects an employee sees in the time sheets. 

1. Is there a report that shows all the projects an employee is assigned to?
2. Is there an easy way to assign an employee to multiple projects at once?

Here are various options you can avail:

1. Run the Project Assignments report from the Reports screen. Add any filters as needed. 
2. To assign an employee to multiple projects, you can use the Manage Access feature (Settings > Access & Permissions). Check CORE Help Center for details.