1. Does a running process always run instructions or does it ever stop? Why
would it stop?

    No, not always. A running programs may be go into a blocked state at any time 
    depending on how the OS's behaviour. For example, if a process needs to wait 
    for data from the hard drive, the OS may context switch to another process to 
    avoid wasting CPU time. 

2. What code runs that will copy the running process into a new running
process?

    fork()

3. If a parent has not yet wait()'ed for their child process but the child
terminates, should the parent still be able to wait for it? Why or why not?
    
    Yes, it should still be able to wait() for it. This is to guarantee that the 
    parent process can always find the child's exit status. If the OS deletes that
    information, then the parent would never know if the child succeeded or encountered
    some kind of error.