1. What's the difference between "向く" and "向かう"?
向く means 'to turn into that direction', whereas "向かう" means 'to turn into that direction, and move forward to that direction or do something actively to that direction'.
・北を向く turn to the north
・北へ向かう go to the north
A. describes merely that the person is looking toward the wall while speaking/talking.
B. implies that there might be somebody on the other side of the wall (the next room in a jail?), or he may be practicing a play of Shakespeare with a picture on the wall as a partner.
2. What's the difference between "やめる" and "とめる"? Do they share the same kanji which is "止める"? As far as I see, "とめる" is used for solid objects such as cars, while "やめる" is used for abstract objects such as speeches, actions. Is that true?
No, I don't think so.
やめる is quit doing something, often permanently and based on one's own will, that one has been doing for a certain amout of time.
The person who quits something is the same person who was doing it.
So, くるまをやめる can mean 'return the driver license and never drive again' rather than 'stop the car'.
・工場は真空管(しんくうかん）の生産をやめた. The factory stopped the production of vacuum tubes (probably for ever).
・工場はLCDの生産をとめた (This implies stopping temporarily. Something happened to the nearby power plant?)
3. Since we have transitive and intransitive verbs of the same meaning, such as "集める" and "集まる", do the sentences "私は古い時計を集める" and "私は古い時計が集まる" mean the same? The second one sounds kinda weird to me.
Yes, weird. Do you think "collecting stamps" is the same as "Stamps get together"?
4. For verbs such as "向く","振り向く", when we are turning into a direction, do we use へ or を? So far the examples I came upon used "を"
Both へ and を can be used. But へ is often used when the *directional object* is somewhat abstract such as West and East.
・うしろを振り向く (look behind)
5. What's the difference between "引き出す"and "出す"?
出す is a very versatile verb. Its basic meaning is to move something from inside to outside to make it visible.
See this for other possible meanings.
引き出す is literarly ひく='pull' (often with some effort) + 出す, thus it means 'to pull out'.
It is also used to express 'to withdraw money from bank'.
・答えを出す get the answer
・答えを引き出す elicit the answer