Hell Tsu,
"to mimic" is to imitate evoking a satirical effect, like making fun of someone's manner or a person's way of talking or walking etc...
It is a specific kind of imitation aiming at showing a person's trait in a funny ,satirical manner to criticize their conduct in general. It is many times done by actors or on stage.
"to imitate" is to copy in general anyone or appear like them in appearance or behavior , it could be also a reproduction of an original object, phenomena such as art does imitate life and reproduce replicas of many natural phenomena .
You friend could imitate you in the way you dress or talk for example ,which could be out of admiration.
"to ape" is just doing everything the other does ,the way they walk ,talk or whatever and all aspects thoughtlessly .
Modeling someone you admire is not the same as aping all they say or do. Aping is usually a silly conduct.
"to simulate" a feeling or an action is when you pretend that you are doing it or feeling it . So you reproduce someone's behavior or look with pretense . It is like fabricating , acting or feigning any action or behavior .