#1: Correct., but I prefer "Lack of sleep can make a teenager tired". The article "A" at the beginning of the sentence is optional. Using the singular is also optional but I think it sounds better. Even though it looks like you are talking about just one teenager, you are not. When you say "a teenager" you are talking about all teenagers.
#2: wrong. You cannot cause an adjective. You cannot cause "happy". You cannot cause "tired". You can only cause a noun or a noun clause like "to become tired". Here are some examples:
His bad driving caused an accident. ("accident" is a noun)
His laziness caused trouble. ("trouble" is a noun)
The noise caused her to lose sleep. ("her" is an indirect object and "to lose sleep" is the direct object and a noun clause)