"I hear it every night!"
Translation:Я чую це щоночі!
It is actually two questions:
What is the case and number of ночі and вечора in the above sentences? They are both in genitive singular. Yes, nominative plural for ніч looks the same as genitive singular: ночі. Genitive is often used to denote possession, so you can think of Я чую це кожної ночі as "I here it (in the course) of every night. By the way, you can use slovnyk.ua to look up word forms (unfortunately it is only in Ukrainian).
Can you say "night" when you mean "evening" in Ukrainian (as in "What are you doing tonight?")? No, normally, there is a strict differentiation between night and evening in Ukrainian. So, if you say Я чую це кожної ночі you mean a time interval from ca. 11PM to ca. 3:59AM, and if you say Я чую це кожного вечора you mean a time interval from ca. 5PM to ca. 10:59PM.