Learn Hardcore Chinese: Tā zài túshūguǎn ménkǒu děng wǒ. - She’s waiting for me at the library entrance.

Here 在 introduces location: 在图书馆门口 (zài túshūguǎn ménkǒu) = “at the library entrance.” It isn’t the progressive marker by itself in this sentence. To stress an ongoing action, you can use:

正在 (zhèngzài): 她正在图书馆门口等我。
Sentence-final 呢 (ne): 她在图书馆门口等我呢。
Aspect marker 着 (zhe): 她在图书馆门口等着我。 Do not say 她在图书馆门口在等我 (double 在). Also, don’t insert 是 (shì) here; 是 links nouns (A is B), not verbs like 等.

Can I say 她等我在图书馆门口 with the place at the end?

That order is generally unnatural. The normal pattern is:

Subject + 在 + place
- Verb (+ Object) → 她在图书馆门口等我。 Alternatives:
她等在图书馆门口。 (uses 等在 with a place complement; you normally drop the object)
Topic-comment emphasis: 在图书馆门口，她等我。 Avoid 她等我在图书馆门口 in ordinary speech.

Should it be 图书馆门口 or 图书馆的门口?

Both are fine.

图书馆门口 (no 的) is very common and flows naturally as a place phrase.
图书馆的门口 explicitly marks possession (“the library’s entrance”) and can feel a bit more formal. When a noun modifies a locational word (门口, 里面, 旁边, 前面, etc.), 的 is often optional.

What’s the nuance of 门口, and how is it different from 门前, 门外, or 入口?

门口 (ménkǒu): the doorway/entrance area (can include just inside or just outside); most common in daily speech.
门前 (ménqián): in front of the door (emphasizes the outside area before the door).
门外 (ménwài) vs 门内 (ménnèi): outside vs inside the door.
入口 (rùkǒu): the designated “entrance” (as on signs). Opposite: 出口 (chūkǒu) “exit.”

Can I add 着, 正在, or 呢 to emphasize “she is (right now) waiting”?

Yes:

她在图书馆门口等着我。 (sustained state)
她正在图书馆门口等我。 (action in progress “right now”)
她在图书馆门口等我呢。 (conversational, ongoing) All are natural; don’t stack them redundantly.

How do I express past, future, or duration with this sentence?

Chinese uses particles and adverbs:

Duration/completed action: 她在图书馆门口等了我一个小时。 (She waited for me for an hour.)
Habitual: 她常常在图书馆门口等我。
Continuous over a span: 她一直在图书馆门口等我。
Future/plan: 她会/要在图书馆门口等我。 To say “until I arrive,” use 等到 + event: 她在图书馆门口等到我来了才走。

How do I negate this correctly?

Not at that place: 她不在图书馆门口。
Not in the process/didn’t wait there: 她没在图书馆门口等我。
Didn’t wait (at all): 她没等我。
Doesn’t wait (habitually/refusal): 她不等我。 Use 不 for general/habitual or refusal; use 没(有) to negate past or ongoing actions.

Any pronunciation or tone-sandhi tips for this sentence?

Pinyin: Tā zài túshūguǎn ménkǒu děng wǒ.

Third-tone sandhi: děng wǒ is pronounced déng wǒ (first 3rd tone becomes 2nd).
图书馆 tones: 2-1-3; 门口 tones: 2-3; 在 is a sharp 4th.

Do 她 and 他 sound the same?

Yes. Both are pronounced tā (first tone). 她 = she/her, 他 = he/him, 它 = it. In speech they’re identical; the distinction appears only in writing.

Is 在 necessary before the place? Could I drop it?

Keep it. 她图书馆门口等我 is ungrammatical. You need 在 to link subject and location:

Correct: 她在图书馆门口等我。 Alternatively, use a place complement with the verb:
她等在图书馆门口。 (but then typically no object like 我)

What’s the difference between 等 and 等待?

等 (děng): neutral, everyday.
等待 (děngdài): more formal/literary or emphatic. Both can mean “wait (for),” but 等 is the default in speech.

If she is inside rather than at the entrance, how would I say it?

Use a different locational word:

她在图书馆里等我。 (inside) Other common options: 前面 (in front), 后面 (behind), 旁边 (beside), 对面 (across from).

How do I say “Wait for me (a moment) at the library entrance”?

Add a short-duration phrase for politeness:

在图书馆门口等我一下/等我一会儿。 You can also specify time:
在图书馆门口等我十分钟。