Hollywood’s most prestigious film awards are also a reminder of the hard work, creativity and courage movie-makers employ to entertain, move and enthrall audiences the world over.

Q1: Is it omitted 'that' between courage and movie-markers?

Q2: Is 'employ to' a phrase that means 'used to' OR just consider 'employ' itself a verb that means 'utilize' and that is followed by a object which is emitted from 'hard work ,creativity and courage'?

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Q1: yes, "that" should really be inserted between "courage" and "movie-makers". It's certainly intelligible without it and probably would go unnoticed.

BTW: your question itself should be "Is 'that' omitted between courage and movie-makers?" (Or even better: "Is the word 'that' omitted...") You've used the pronoun 'it' before one could infer what 'it' refers to.

Q2: I'm not sure I understand this one. "Employ" just means to 'utilize' as you suggest. The things (direct objects?) that we're utilizing are "hard work, creativity and courage" and our goal is "to entertain, (to) move and (to) enthrall audiences"