Learn Hardcore Chinese: tā zhèngzài tīng tā shuō.

正在 marks an action in progress right now; it’s the closest to English -ing with a strong “right now” feel: 她正在听他说.
在 alone also marks the progressive, but is a bit less emphatic: 她在听他说.
正 alone also works and often sounds a bit more concise/literary: 她正听他说.
With none of them, context decides; it can read as habitual or a simple statement: 她听他说.

Can I omit 说 and just say 听他?

Be careful. 听他 by itself often means “to obey him/listen to his advice.” To make it clear you mean “listen to him speak,” add a speaking verb:

她在听他说话 or 她在听他讲话 or keep the original 她正在听他说.
Note: 我听他的 commonly means “I’ll do as he says,” not “I’m listening to him speak.”

What’s the difference among 说, 说话, and 讲话?

说 = say/speak; often takes content (explicit or implied). With 听, 听他说 is common and natural.
说话 = speak/talk (general act of speaking); very common in Mainland: 他在说话, 听他说话.
讲话 = speak/give a talk; common in Taiwan and in formal contexts: 他在讲话, 听他讲话. All three can work here; 说话 feels very neutral for “speaking,” 讲话 may sound a bit more formal/regional, and 说 is concise, especially when content is implied.

Why is 他 in the middle? Why not say 听说他?

This is a common serial-verb/pivot structure: 听 + 他 + 说 = “listen to him speak.” The object of the first verb (他 as the object of 听) also acts as the subject of the second verb (说).

Other examples: 看他写字 (watch him write), 听她唱歌 (listen to her sing), 让他来 (let him come).

Is 她听他说 (without 在/正在/正) also correct?

Yes, but it’s more context-dependent:

As an ongoing action right now, it’s safer to mark the progressive: 她在/正在听他说.
她听他说 on its own is often read as habitual or generic (she listens to what he says), unless context makes it clearly ongoing.

Can I add 呢 to emphasize the ongoing action?

Yes. Sentence-final 呢 adds an “it’s in progress” feel:

她正在听他说呢。
她在听他说呢。

How do I negate this? Is 不在 correct?

To negate an ongoing action, use 没(在): 她没在听他说 or 她没有在听他说.
她不听他说 = she doesn’t listen to him (habitually/refuses to).
Avoid 她不在听他说, because 不在 usually means “not at (a place).”

How do I talk about past or future with this sentence?

Mandarin uses time words/modals rather than tense changes:

Past/ongoing then: 昨天她在听他说。 / 那时候她正在听他说。
Future: 她明天会听他说。 / 她明天要听他说。

What’s the difference between 听, 听见, and 听到?

听 = to listen/hear (neutral/in-progress): 我在听他说.
听见 / 听到 = to hear (achievement/result reached): 我听到他在说话 (I heard him speaking). Don’t mix progressive with a result verb like 正在听到 (unnatural).

Do I need to add 什么 after 说 to mean “what he’s saying”?

Add it when you want to highlight unknown/queried content:

她正在听他说什么。 (She is listening to what he is saying.) You can also follow 说 with a clause when the content is known:
她在听他说他的经历。

Pronunciation and tones?

她 tā (1st), 正在 zhèngzài (4th-4th), 听 tīng (1st), 他 tā (1st), 说 shuō (1st).
说 here is always first tone shuō. Both 他/她 are pronounced tā in speech.

In speech, how do I tell 他 and 她 apart?

You don’t; both are tā. Context tells you. In writing:

他 = he/him (also historically generic),
她 = she/her,
它 = it (things/animals; 牠 is also used for animals, especially in Taiwan).

Can I replace 说 with specific activities like telling a story or singing?

Yes. The pattern is very productive:

她正在听他讲故事。 (listen to him tell a story)
她在听她唱歌。 (listen to her sing)
她在听他发言。 (listen to his speech/remarks)

Is 她正在听他在说话 okay?

Grammatically possible, but less common/redundant. Native speakers usually say:

她正在听他说话。 Use 在 inside the embedded clause when you need to highlight that ongoing nature for a reason, e.g. 我听他在跟她说话。

Does 听说 mean the same as 听他说?

No.

听说 = “it’s said/I’ve heard (from unspecified sources)”: 我听说他要走。
听他说 = “I heard him say / according to him”: 听他说，他要走。 Your sentence uses 听他说, not the fixed expression 听说.

How do I say this politely to a teacher or elder?

Use the polite second person 您:

她正在听您说。
我在听您说。