What does 冲 in 大老鼠冲着胆小先生嚷嚷 mean?
冲着 in this case means towards. 'Toward' itself can be regarded as an action, even though it may not be in the literal sense of 'movement'.
冲 indeed generally means physically rush. However, in case of 冲着, it is never physical, but directional. Compared to 对着, 冲着 implies a sense of urgency and harshness.
A simple example:
我对他喊 vs 我冲他喊
Both means 'I shout to him'. The later can be used in scenarios such as calling for help or shouting angrily. The former is neutral by itself.
In your specific case, 大老鼠冲着胆小先生嚷嚷. 嚷嚷 is complaining/scolding/make a fuss. 冲着 here is used to emphasize the negativity of 嚷嚷.
Maybe the below example would help you understand why 冲 (physically rush) can convey a negative sense in conversation.
When we want to describe someone's way of talking to others is aggressive, we may say that they 说话(the way they talk)很(very)冲.
Rush is an action about advancing quickly and often with little regard of consequence or surrounding. It's not hard to connect the dots and figure out why it could be compared to aggressiveness