To use these words in sentences, you need to know what they mean and what part of speech they are.
Smirking and carousing are both verbs in the progressive or continuous tenses. (They can be other things like gerunds, but we will use them both as progressive tense verbs in this answer.) That means that they are being done on a continuing basis. To smirk is to smile in a smug or conceited way. People smirk when they feel like they have gotten away with something or because they feel like they are better than the person they are smiling at. So write a sentence where you say that someone is smirking and you explain why they are. To carouse is to go out, drink a lot and act rowdy. Write a sentence where someone is doing that.
Orneriness is a noun. It is the state of being hard to control, disagreeable, or easily angered. Write a sentence where you say that someone “showed his orneriness when he” and then say what he did that showed that he was hard to control, disagreeable, or quick to get angry. An insomniac is a person who has insomnia, which is a problem in which you find it hard to sleep. Write a sentence about an insomniac where you say “Because she couldn’t sleep, the insomniac…” and then go on to say what she did instead of sleeping.