Learn Hardcore Chinese: tā zhàn zhe děng gōngjiāochē, wǒ zuò zhe kàn shū. - He is standing and waiting for the bus; I am sitting and reading.

It marks a continuous state that accompanies another action. In 站着等, 坐着看, the verb before 着 (standing, sitting) is the ongoing posture/manner, and the verb after it (waiting, reading) is the main action.

Why is 着 used after 站 and 坐, but not after 等 or 看 here?

Because the structure is “posture/manner verb + 着 + main action.”

站着等 = wait while staying in a standing state.
坐着看 = read while staying in a seated state.
If you put 着 on 等, you get 等着, which means “be waiting (for something to happen),” often used as 等着他来 / 等着车来.
If you put 着 on 看 (i.e., 看着), it usually means “looking at/staring at,” not “reading,” so it would change the meaning.

Is 着 (zhe) the same as 在/正在 for the progressive “be doing” in English?

No.

在/正在 marks the main event as in progress: 他正在等公交车 (He is waiting for the bus).
着 marks a sustained state/posture: 他站着等公交车 (He waits, standing).
You can combine them for emphasis: 他正站着等公交车 or 他站着在等公交车.

Can I say 他在站着等公交车 or 他站着在等公交车?

他站着在等公交车 is acceptable and emphasizes both the posture and the ongoing action; it’s a bit wordy but natural.
他在站着等公交车 is less common; prefer putting 在/正在 before the main action (等) rather than before the whole compound starting with 站着.
A very natural option is 他正站着等公交车.

Can I omit 着 and just say 他站等公交车, 我坐看书?

No in modern colloquial Chinese. Without 着, 站/坐 are interpreted as actions (“to stand up,” “to sit down”) rather than a continuing posture. Use 站着等, 坐着看 to mean “wait while standing,” “read while sitting.”

Could I use 了 instead of 着?

No. 了 marks completion/perfective, not a continuing state. 他站了等公交车 is ungrammatical. If you want duration, you can say:

他站着等了很久 (He stood waiting for a long time).
Or split it: 他站了很久，等公交车。

Can I use 一边…一边… here?

Use 一边…一边… for two simultaneous actions by the same subject (e.g., 他一边等公交车，一边看手机). It’s not natural to pair a posture with an action using 一边 (e.g., 一边站着，一边等公交车) because the posture is better treated as background with 着.

Why is the word order 站着等, not 等站着?

Chinese often puts the manner/posture first: [manner/posture] + 着 + [main action]. So 站着等 and 坐着看 are the standard orders. If you really want to highlight the time frame, you could say 等公交车的时候，他站着.

Why is there a comma instead of 和 (and) between the two clauses?

Chinese commonly uses a comma to coordinate two independent clauses with different subjects. Using 和 to join whole clauses is uncommon. You could also say 他站着等公交车，而我坐着看书 to highlight contrast.

How do I pronounce 着 here, and how is it different from its other readings?

Here it’s the neutral-tone particle zhe (durative state). The character 着 also has:

zháo: result, as in 睡着 (shuì zháo, fall asleep).
zhuó: a literary reading in words like 着装 (zhuózhuāng, attire).
Don’t pronounce the particle here as zháo or zhuó.

Does 看书 always mean “read a book”?

Usually it means “read (books/printed material)” in general, not necessarily one specific book. It contrasts with, say, 看手机 (look at the phone) or 看报 (read the newspaper). For a specific book, add a measure word: 看一本书.

What’s the difference between 看书 and 读书?

看书: to read (books/printed material), a concrete activity.
读书: to read/study; often means “to be in school” in context (e.g., 他在北京读书 = He studies/goes to school in Beijing). In some contexts 读书 can also mean “to read books,” but 看书 is the default for the activity of reading.

Is 公交车 the same as 公共汽车/公车/巴士?

Mainland: 公交车 or 公共汽车 are standard; 公交 is a common short form (colloquial).
Taiwan: 公车 is common.
Hong Kong and elsewhere: 巴士 is common.
All mean “bus,” but use the form natural to the region.

Do I need measure words for 书 or 公交车 here?

Not in this sentence. You add measure words when you specify quantity:

一本书 (one book), 看一本书.
一辆公交车 (one bus), 等一辆公交车 (rare—people usually just say 等公交车 unless specifying which/one).

How do I negate these?

Negate the ongoing action: 他没在等公交车 / 他不等公交车 (he isn’t/doesn’t wait for the bus).
Negate the posture-plus-action idea: 他不站着等公交车 (he doesn’t wait standing).
For the second clause: 我不坐着看书 (I don’t read while sitting) or simply 我没在看书 (I’m not reading).

How do I turn this into a yes–no question or use 呢?

Yes–no with 吗: 他站着等公交车吗？ 你坐着看书吗？
Using 呢 to emphasize ongoingness: 他站着等公交车呢。你呢？
Or progressive: 他在等公交车吗？你在看书吗？

How do I add a location naturally?

Use 站在/坐在 + place:

他站在车站等公交车。
我坐在椅子上看书。
Avoid 站着在…; prefer 站在… or keep 站着 and add 在 before the main action: 他站着在车站等公交车.

Is 他站着等公交车 different from just 他等公交车?

Yes. 他等公交车 just states the event (he’s waiting for the bus). 他站着等公交车 adds the manner/posture, making the scene more specific (he’s waiting while standing).

Can I say 他站着等着公交车?

It’s better to give 等着 a following event or target:

Natural: 他站着等着公交车来。 (He is standing, waiting for the bus to come.)
Without 来/到/人/事, 等着公交车 sounds incomplete in many contexts. If you don’t need that nuance, just use 等公交车.