Does a multitasking system involve a mouse cursor to make the user able to interact with more than one task/process at a time?
You don't need a mouse to have a multitasking system. The Wikipedia article on multitasking has some history of multitasking systems; they're a lot older than window environments and mice. The first multitasking systems ran batch jobs: you submit a task (by loading up a deck of punched cards, for example) and wait for it to finish; there could be multiple tasks in progress at any given time.
Later systems had user interaction through a command line; for example, in a purely textual unix user interface, you can use job control to run commands in the background, and control which program you get to interact with.
Even in a typical window environment, the application that has the focus (i.e. the application that you type into) isn't the only one that can get CPU time. A window environment on a multitasking operating system lets you switch to another window while an application is computing something. Additionally pretty much any multitasking system has a bunch of tasks ready in the background, only running when some event happens (hardware event, packet received over the network, timer, …). So even when there are windows and a mouse, there's no particular relationship between them and multitasking.