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 Assignment report. You can run this report directly from the Project Assignments screen or from the Reports screen in case you need to use any filters. 
2. To assign an employee to multiple projects, you can use the Employee Control feature (main company menu > Settings). Check Core Help Center for details.