Learn Hardcore Chinese: nǐ wèishénme hái zài gōngsī?

It means "still/yet": continuation of a state. In 你为什么还在公司？, 还 modifies 在, giving "still be (at)". If you meant "also", you'd typically use 也: 你也在公司吗？

Why is there no 是? Why not 你为什么是还在公司? or 你为什么是在公司?

In Chinese, location is expressed with the verb 在 ("to be located at"), not with 是. So the predicate is 在公司. 是 can appear only for special contrastive focus, e.g., 他不是在家，他是在公司 (He is not at home; it's at the office that he is), but it's not needed in the neutral question.

Can I put 吗 at the end: 你为什么还在公司吗？

No. You don't use 吗 with wh-questions like 为什么. Use 吗 only for yes–no questions, e.g., 你还在公司吗？ (Are you still at the office?).

Where does 还 go in the sentence?

Place 还 before the verb or adjective it modifies. Here it goes before 在: 还在. With other verbs: 你为什么还没走？, 他还工作呢. You wouldn’t put 还 after 在 or before the subject.

Why use 为什么 and not 怎么? What's the difference between 为什么还在公司？ and 怎么还在公司？

为什么 asks for a reason/explanation ("why?").
怎么 here means "how come", often expressing surprise or mild criticism.
So 你为什么还在公司？ is a neutral "why", while 你怎么还在公司？ feels more like "How come you're still at the office?!"

Is 在 a verb or a preposition here?

Here 在 functions as the main verb "to be (located) at/in". The predicate is 在公司 ("be at the company"). In other sentences it can introduce a location phrase before another verb, e.g., 他在公司工作 ("He works at a company"), where 在公司 is a coverb phrase modifying 工作.

Do I need to add 里 after 公司? What's the difference between 在公司 and 在公司里?

Both are fine. 在公司 is the default and means "at the office/company (premises)". 在公司里 adds the sense of "inside the company building/inside the office". In everyday speech, 在公司 is more common.

Does 公司 mean "company" or "office"? Should it be 办公室 instead?

In this context 公司 often stands for "the workplace/the office (i.e., your company)". 办公室 is the office room.

在公司 = at the company/at work (broader).
在办公室 = in the office room (more specific).

What’s the difference between 还在 and 还是在? Can I say 你为什么还是在公司？

Both can mean "still at", but:

还在 is the default, neutral "still".
还是在 adds contrast or persistence against expectation: "still (after all that)".
你为什么还是在公司？ is possible but tends to sound more emphatic/exasperated. For a neutral question, use 还在.

How is 还 different from 也, 再, and 又?

还 (hái) = still/yet; also "also" in some contexts. Here it's "still".
也 (yě) = also/too (adds another item/person).
再 (zài) = again (future/repetition), "more" when with numbers: 再来一个.
又 (yòu) = again (past/repeated/recurring).
You would not say 再在公司 here; 又在公司 would mean "at the office again".

Pronunciation check: How do I say this sentence? And does 还 ever read huán?

你为什么还在公司？ nǐ wèishénme hái zài gōngsī
Tones: 你(3), 为(4), 什(2), 么(neutral), 还(2), 在(4), 公(1), 司(1).
还 is pronounced hái when it means "still/also". It’s huán only when it means "to return (something)", e.g., 还书 huán shū.

Why not move 为什么 to the very front: 为什么你还在公司？

You can, and it's grammatical, but everyday speech prefers 你为什么…. Putting 为什么 first can sound slightly more formal or emphatic on the reason.

Can I add a softening particle at the end, like 呢 or 啊?

Yes.

你为什么还在公司呢？ sounds gentler/curious.
你怎么还在公司啊？ is colloquial and expressive.
Avoid 吗 with 为什么.

How do negatives work here? What's the difference between 不在 and 没在?

For present location, use 不在: 他现在不在公司 (He is not at the office now).
没在 negates past occurrence or duration, e.g., 我去年没在这家公司工作 (I didn’t work at this company last year). In casual speech some people say 他没在公司 for "he's not at the office", but 不在 is the standard for current location.

Where do time words go? For example, "now" or "so late".

Time expressions usually go after the subject and before the verb:

你现在为什么还在公司？
你为什么现在还在公司？
Both are acceptable; putting 现在 right after the subject is very common.

Can I drop 你 if it's clear who I'm talking to?

Yes. In context you can say 为什么还在公司？ or text 还在公司吗？ Pronouns are often omitted when obvious.

Do I need a classifier with 公司 here? What about when I specify which company?

No classifier is needed in the location phrase 在公司. When specifying a particular company, use the classifier 家: 在那家公司, 在一家科技公司.

Are there synonyms for 为什么 that change the tone or register?

Yes:

为何 (wèihé): formal/literary.
为啥 (wéishá): colloquial (north).
干嘛/干吗 (gànmá): very colloquial "what for/why (are you doing that)".
So you could hear 你为啥还在公司？ or 你干嘛还在公司？

If I want to say "Are you still working (there)?", how would I phrase it?

Use 上班/加班:

你还在公司上班吗？ Are you still working at that company?
你怎么还在公司加班？ How come you’re still working overtime at the office?

How would I answer this question in Chinese?

Use 因为… for "because…":

因为要开会，所以我还在公司。
因为堵车，我还没走。
A short reply is fine too: 还在加班呢。