Learn Hardcore Chinese: qǐng mànmàn de shuō, wǒ xiǎng tīng dǒng. - Please speak slowly; I want to understand.

的: links modifiers to nouns (adjectival). Example: 慢的车 “a slow car.”
地: turns adjectives/adjectival phrases into adverbs to modify verbs (manner). Example: 慢慢地说 “speak slowly.”
得: follows a verb and introduces a complement of degree/result/manner. Example: 说得很慢 “speak very slowly.”

So here 慢慢地说 uses 地 because “slowly” modifies the verb “say/speak.”

What does the reduplication 慢慢 add compared with just 慢 or 慢一点?

慢 (slow) is the base adjective.
慢慢 is a common reduplication that means “slowly, in a slow manner,” often emphasizing the process or gentleness (also “take your time” in some contexts).
慢一点/慢一点儿 means “a bit slower” and quantifies the degree of slowness. Examples:
慢慢地说 = “Speak slowly (in general/throughout).”
说慢一点 = “Speak a bit slower (than you are now).”

Do I have to include 地? Is 慢慢说 acceptable?

Yes, 慢慢说 is very common in speech and perfectly acceptable. Adding 地 (慢慢地说) is a bit more formal or textbook-like, and more common in writing. Both are correct.

Is 请 necessary? How else can I ask politely?

请 softens an imperative, but you can omit it. Other natural options, often even politer:

能/可以说慢一点儿吗？ “Could you speak a bit slower?”
麻烦你说慢一点儿。 “Could I trouble you to speak a bit slower?”
说慢一点儿，好吗？ “Speak a bit slower, okay?”
With a subject: 请你慢慢地说。 (adding 你 makes the addressee explicit)

What exactly does 听懂 mean? How is it different from 听到/听见/听清楚/听明白?

听懂: to understand (by listening). Focus on comprehension.
听到/听见: to hear (perceive sound). Focus on perception.
听清楚: to hear clearly (no ambiguity in sound). Focus on clarity of hearing.
听明白: to understand clearly (similar to 听懂, often slightly more explicit about clarity). So you ask someone to speak slowly so you can 听懂 (understand), not just 听到 (hear).

Should it be 听得懂 or 听懂 here? What’s the difference?

听懂 states an achieved result for a specific event: “(I) understood (by listening).”
听得懂/听不懂 (potential complement) expresses ability or possibility: “can/can’t understand (by listening).” In your sentence, 我想听懂 = “I want to end up understanding.” You could also say 我想听得懂 = “I want to be able to understand.” Both are acceptable; the first leans toward the result of this conversation, the second toward general ability.

Why use 想 here? Could I use 要, 能, or 希望 instead?

想 = “would like to; want to” (polite/mild).
要 = “want to; going to” (firmer/stronger).
能/可以 = “be able to/can; be permitted to.” With 听懂: 我希望能听懂 “I hope I can understand.”
希望 = “hope.” Softer, expresses wish. Politeness/natural alternatives:
请慢慢地说，我希望能听懂。
请说慢一点儿，我怕听不懂。 (I’m afraid I won’t understand.)

Can I say 请说慢一点(儿) instead? Is it more natural?

Yes, 请说慢一点(儿) is extremely common and arguably more idiomatic than 请慢慢地说 in casual conversation. You’ll also hear:

请说得慢一点(儿)。 (using 得)
Without 请: 说慢一点(儿)，谢谢。

What’s the difference between 慢慢地说 and 说得慢?

Two different structures to express manner:

[Adverbial] + 地 + [Verb]: 慢慢地说 “speak slowly” (pre-verb adverb).
[Verb] + 得 + [Descriptor]: 说得慢 “speak slowly” (post-verb complement of degree). They’re near-equivalents. With quantification you often see the 得-pattern: 说得慢一点(儿).

Where would I put the object if I want to say what I’m trying to understand?

Place it after the verb or after the resultative verb:

请慢慢地说，我想听懂你说的话。 “... I want to understand what you’re saying.”
Other objects: 我想听懂这段/内容/中文。 You can also make the request more explicit:
请你慢慢地说，这样我才能听懂你说的话。

How is 地 pronounced here? Any tone changes in the sentence?

As a structural particle, 地 is pronounced neutral-tone de (not dì).
Tone sandhi: in 我想 (both 3rd tone), the first 3rd tone becomes a rising 2nd tone in fluent speech: wó xiǎng.
Others: 请(qǐng) 3rd, 慢(màn) 4th, 说(shuō) 1st, 我(wǒ) 3rd→2nd before 想, 听(tīng) 1st, 懂(dǒng) 3rd.

Is 我想听懂 a natural thing to say? What might natives say instead?

It’s acceptable and understood. Common alternatives you’ll hear:

我还听不太懂。 “I still don’t quite understand.”
我怕听不懂。 “I’m afraid I won’t understand.”
我想听得懂你说的话。 “I want to be able to understand what you say.”
我希望能听懂。 “I hope I can understand.”

Can I drop pronouns like 我 or 你 in this sentence?

Yes, if they’re clear from context. Mandarin often omits subjects when understood:

请慢慢地说，(我)想听懂。
请(你)慢慢地说。 In a first-time request to a stranger, keeping 你/我 can sound clearer/politer: 请你慢慢地说，我想听懂。

Can I use 讲 instead of 说?

Often yes. 讲(jiǎng) overlaps with 说, especially for “to speak/explain/talk about.”

请讲慢一点(儿)。 is natural. Nuance:
说 is the safest general word for “say/speak.”
讲 can lean toward “explain/talk about” (讲道理, 讲课). Regional preferences vary (讲 is very common in many regions, including Taiwan).

Why is there just a comma between the two parts? Could I use 因为/所以?

The comma separates two closely linked clauses: a request and a reason. You could add connectors, but it’s not necessary:

因为我想听懂，请慢慢地说。 (OK, slightly bookish)
请慢慢地说，这样我才听得懂。 (very natural, using 这样…才 to link cause → effect)

What about 慢点(儿)? Is that different from 慢一点(儿)?

They’re the same in meaning (“a bit slower”). 点儿 is the northern/colloquial variant of 一点儿:

Mainland north: 慢点儿
Elsewhere/neutral: 慢一点 (no -er) All of these are fine: 慢点, 慢点儿, 慢一点, 慢一点儿.