You can in spoken Arabic and it is used very often in certain countries like Lebanon. The makers of the course aren't trying to teach pure MSA here (Modern Standard Arabic). They're trying to strike a balance between MSA and spoken language. As they mentioned in an article linked during the course launch:
we’re teaching a less-formal, spoken version of Modern Standard Arabic — not the version that would appear in poetry or formal news broadcasts, but instead the version that would be used once a newscaster stopped reading from their script and started talking to their interviewee. It’s a version of the language that can be used in a formal conversation, but one that also can be used with the widest range of Arabic speakers.
You can read their full article here https://making.duolingo.com/what-makes-arabic-hard-and-why-that-shouldnt-stop-you-from-learning-it
Sleepy and tired are most definitely different, both in English and Arabic. Sleepy = نغسان Tired = تعبان
You can be tired after a long run, but that doesn't mean that you want to sleep. Sleepy means that you feel like sleeping. It's different. I presume from your name that your native language is Dutch, and in Dutch (as well as German), people often say they are tired when they want to go to sleep, not so in English and Arabic.
The (1) "you" is from وأنتِ which means "and you?" (refers to 2nd person feminine singular) and the (2) "you" is from the ك (another "you" form) of كيفِك which means "how are you?".
Note: كيفِك is not Standard Arabic (but Levantine). In Standard, it is كَيْفَ حالُكِ which كِ refers to you (2nd person feminine singular).