Note that "What for?" is informal/conversational.

"What Purposeful Reason?"

"What for" denotes a purposeful reason, while "Why?" can be used for causes, reasons, or explanations.

• Statement 1: "I am going to work now."
• Why? (Explanation): "Because it's time for me to leave."

• Why? OR What for? (Purpose): "To make money."

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• Statement 2: "Things fall."

• Why (explanation): "Because of gravity..."
• Why (Purpose): "Because of gravity." / "No, I mean, why is it like that? Why is there gravity?" / "Who knows? That's just the way it is. God? Quantum Multiverse? That may not be answerable."

• What for (purpose only!): "Who knows? God? Quantum Multiverse? That may not be answerable."

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• Statement 3: "I exist."

• Why (Explanation/Cause): "Because your parents had sex, your mother got pregnant, and you were born."
• Why OR What for? (Purpose): "In order to propagate the species." OR "For God's mysterious reason." OR "Nobody knows." OR "There is no purpose; it's a quantum multiverse."

Variations can be created by sentences like "What is $NounPhrase$ for?" For example, if one is in a new car with a salesman, one can ask "What is that button for?" In this case, the question is asking for the functional purpose of the button. Also, "What is math good for?" is asking for suitability for a purpose.

"What for!?" -- Emphasis / Surprise / Suspicion

"What for" can lend itself to more emotion than a simple "Why", adding surprise, suspicion, or just more emphasis. In this case, "what for" can be asking for an explanation just like "why":

• Statement 4: "I am going to work now."
• What for!? You just got home! (Surprise/Explanation): / "Because I left my wallet there."

The emotion behind "What for?" can also be suspicion or interrogation. There can be a bit of eye squinting or head tilting:

"Can I ask you your name?"
"Ummm. What for?" (Suspicion)

"For what?" -- Identifying ambiguity.

"What for" can also be used to mean "For what?" in order to identify an ambiguous reference. The emphasis is on replacing the "what" with the named item rather than replacing the "for" with the named reason:

1. "I'm looking for something." / "For what?" / "For my glasses."
2. "I'm looking for something." / "What for?" / "My glasses." (Same as "For what")
3. "What are you looking for?" / "My glasses."

Here's another one:

• "I need some help." / "What for?" / "To reach that top shelf." / "Ok, what for?" / "That coffee mug."

Both "what for" questions could be asking for purposeful reason. For example, in the 2nd case, the reason could have been to clean the shelf. But the answer given in the 2nd case was an identification of the object desired.

Also, "What am I responsible for?" is asking for identification of responsibilities. It is like "For what am I responsible?", but the fronted what sounds more natural.

In their closest meanings, "still" is used for a situation that is ongoing or unchanged at any point in time, or regardless of any factor, stated or otherwise:

She still loves you. If you lose all your money, she will still love you. Even when you were abusive, she still loved you.

"yet" has a similar interpretation, but describes the state of a something, within a process, at any given time within a progression, up to that point in a process:

He is yet a child. -or- He is a child yet.

He's still a child.

In another usage, it is closer to "already." In questions, it asks whether something has happened "already" or up to the time of questioning, and additionally implies that the addressee is expected or required to do it:

Have you called your mother yet?

Here the speaker possibly commanded or just expects the addressee call his mother, or was informed of the his intention to call his mother, and is questioning whether he has done it. "Already" doesn't have this force - it merely questions the completion of the action logistically:

Have you already called your mother? I wanted to say hello to her. (=Am I too late?)

Note that the idiom "have yet to" is used with the above meaning to express that something expected still hasn't been done:

He has yet to call his mother. (= He's supposed to do it, but hasn't done it.)

I have yet to see that movie. (= I've been meaning to see it, but haven't gotten around to it.)

"Not yet" - or "yet" in any negative context - is the opposite of "already:"

My father already sent me a birthday present.

My father hasn't sent me a birthday present yet.

Finally, "yet" can function as a conjunction like "but, however" that emphasizes a result obtained or a situation that arises despite certain factor(s):

He's only been speaking English for a year, yet he sounds just like an American.

I've asked you a million times not to lock the door, yet every time I come home, it's locked.

Note that "still" can replace "yet" here with the same meaning.