Is there an idiomatic or slang word meaning a submissive person who does everything others order him to do and never complains even when they should, and is ,in a sense, controlled by someone else.
one that submits without protest to abuse or indignities.
An even better word is kowtower
someone who agrees too easily or eagerly to do what someone else wants you to do : to obey someone with power in a way that seems weak.
But the best one is he/she is my bitch
A submissive person (often female), who does what others want.
In the sense that an alpha is the dominant member of a group (as in alpha male), and a beta the alpha's lackey:
Omega … is an antonym used to refer to the lowest caste of the hierarchical society. Omega animals are subordinate to all others in the community, and are expected by others in the group to remain submissive to everyone. Omega animals may also be used as communal scapegoats or outlets for frustration, or given the lowest priority when distributing food.
Admittedly, this usage is rare, though it should be clear to those familiar with this sense of alpha or beta (and that usage is common enough).
You could also consider Yes-Man, someone who automatically agrees with everything their superior(s) tell(s) them and only tells them what they want to hear.