Generally, "getting bugged" is something someone else does to you, rather than an emotion you feel like annoyance.
For instance, we can say "I am feeling annoyed" but we wouldn't say I am feeling bugged".
Most of these words gain their meaning contextually. For instance, if you said it bothers you when somebody leaves the dishes out, everybody would understand it's not as severe as something more serious that bothers you emotionally.
I think that "bother" is the strongest word here, because it suggests a feeling less precise and perhaps overwhelming. Feelings like annoyance and frustration tend to pass quickly and be caused by a specific event, but that's not always the case with "bother".