I’m sure you’ve heard the phrase “never end a sentence with a preposition”, which is why you don’t say “who are you going with” but rather “with whom are you going” (it’s also how you know whether to use who or whom…if there’s a preposition involved with the who, it becomes whom).

anti- and counter- are prefixes, but over, under are prepositions. Anything that tells you the relation of one thing to another is a preposition. In “overacting”, “over-” is both a prefix and a preposition.

There is no name for the words that come before, but the combination of words is called a modifier, because it modifies the noun.

So in “counter-terrorism efforts”, “counter-terrorism” is the modifier and “efforts” is the noun.

It would be incorrect to use “counter-terrorism” without a noun behind it. If you did, the hyphen would disappear and it would simply be “He is working to counter terrorism.”