Sentence 1- 'Actions' here means 'tasks'. So you can take an action (do one task), or take actions(do more than one task) Sentence 2- 'Action' here means movement, energy, activity. You could not use 'actions' in this sentence because action has no plural form when used in this way.

Where did you find these sentences? In my opinion, the first sentence should not be plural. It sounds awkward and unnatural to my American English acquainted ear. Even if many actions can and will be taken the expression is: We should take action to deal with the problem. In the second sentence it should not be plural either. In this case, action describes a type of scene. It could read: Children like to see films full of action scenes. The noun it describes is understood.

I agree with Susanne. You could also look at the first one as a 'set phrase' in American English.

"We should take actionS to deal with the problem." For the first sentence, there are MANY actions you can take or do to solve the problem, that's why you use the plural form. "Children all like to see films full of actioN." For the second sentence, you're already using the word "FULL", which can mean MANY/A LOT, and in this case, action is used in a GENERAL way, meaning all/different kinds of action and not just one.