1)So if somebody was choking (maybe on a yummy piece of chicken), which word would you use?
We use this expression "사레들다." when you get something caught in your windpipe and can't help coughing.
2)What if someone was nervous before giving a performance (such as a speech or singing a song) and cleared their throat? Which word would you use?
In that situation, we usually use this expression "목을 가다듬다."
Either 목(청)을 가다듬다 or 헛기침(을_)하다 is okay?
"헛기침 하다" is possible but we usually prefer using the former expression(목(청)을 가다듬다) or 목청을 풀다 when you clear your though before singing a song or performing a speech.
3)And what is the difference between 목 and 목청?
목 means a throat while 목청 means vocal cords. They have different meanings but we can think "목" contains the meaning '목청". So you can choose either of them, when you face the moment you use that expression. However, we usually prefer using the expression containing the word "목", not "목청".
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There is not enough room for explaining this, so I explained it to the new answer post.
I hope this helps. : )