Learn Hardcore Chinese: bùguǎn wèntí duō nán, tā dōu xiǎng shìshi shuō yíxià zìjǐ de xiǎngfǎ. - No matter how difficult the question is, he wants to try expressing his own ideas.

The 不管 … 都 … structure tells you: the result (he wants to try expressing his ideas) does not change even if the condition (question difficulty) changes.

Why is 多 in 问题多难? Why not say 问题很难 or add 吗 like a question?

Here, 多 is not “many/much”; it is the “how …” used with adjectives:

多 + adjective = “how + adjective”

For example:

你不知道这件事多麻烦。
= You don’t know how troublesome this matter is.
我没想到这么多。 (different use: so much / so many)

In our sentence:

问题多难 literally = how difficult the question is
It’s part of the “no matter how …” structure: 不管 + 多 + adj

Compare:

不管问题多难 = No matter how difficult the question is.
不管问题很难 – not correct in this pattern.
问题难吗？ – a normal yes/no question, completely different function.

Typical pattern:

不管多 + adj，(都/也) + result
e.g. 不管多累，我都要完成。 = No matter how tired I am, I must finish.

Why is 都 used when the subject is just one person (他)? Doesn’t 都 mean “all”?

都 can mean “all”, but in this type of structure it has a different function.

In 不管……都……, 都:

links the “no matter what” part with the result
has a meaning like “still / in any case / in every situation” rather than “all people / all things”

So:

不管问题多难，他都想试试……
- = No matter how difficult the question is, he still wants to try…
- = In every possible degree of difficulty, he wants to try.

In “no matter” sentences, you almost always see 都 or 也 after the subject:

不管天气怎么样，我们都去。
无论谁来，我也不见。

Here 都 is about “for all cases”, not about plural people.

Could I replace 不管 with 无论 or 不论? Are there differences?

Yes, they are very close in meaning and structure:

不管问题多难，他都想试试说一下自己的想法。
无论问题多难，他都想试试说一下自己的想法。
不论问题多难，他都想试试说一下自己的想法。

All are grammatical and natural.

Differences (small and mostly about style):

不管: most common in spoken Chinese; neutral, very natural.
无论: slightly more formal / written; often appears in written texts, speeches.
不论: similar to 无论 but used less frequently in everyday speech.

For everyday conversation, 不管 is often the best choice.

What’s the difference between 试试, 试一试, 试试看, and 试着?

All relate to “trying”, but usage and nuance differ:

试试
- Verb-reduplication.
- Very common, casual: have a try / try it out a bit.
- Here: 想试试 = wants to give it a try.
试一试
- Slightly longer, a bit more neutral/formal than 试试, but the meaning is almost the same.
- You could say: 他都想试一试说一下自己的想法。
试试看
- Literally “try and have a look”.
- Very colloquial; slightly more “encouraging / exploratory”.
- Example: 你试试看嘛。 = Come on, (just) try it.
试着 + verb
- Means “try to do (something)” (focus on the attempt to perform another action).
- Example:
  - 他想试着说一下自己的想法。
    = He wants to try to express his ideas.

In your sentence, all of these are possible (with small nuance differences):

他都想试试说一下自己的想法。
他都想试一试说一下自己的想法。
他都想试试看，说一下自己的想法。 (usually with a pause/comma)
他都想试着说一下自己的想法。

Why is there 说一下 instead of just 说? What does 一下 add?

一下 after a verb can:

Indicate a short / brief action.
Soften the tone, making it more polite / less forceful.

So:

说一下 ≈ say (it) briefly / just say (something)
It sounds more casual, friendly, or modest than bare 说.

Compare:

他说自己的想法。
= He states his opinions (sounds more firm / factual).
他说一下自己的想法。
= He just briefly explains/gives his ideas (softer, less heavy).

In the original sentence, 试试 + 说一下 together emphasize:

he only wants to give it a try, and
he’ll just say a bit / try to express his ideas briefly.

What’s the function of 自己 in 说一下自己的想法? Could I drop it?

自己 is a reflexive pronoun: “self / oneself”.

自己的想法 = one’s own ideas/thoughts/opinions.

Here it emphasizes that:

He wants to express his own thoughts, not repeat others’.

If you say:

说一下想法 – grammatically okay, but vague; whose ideas?
说一下他的想法 – his ideas (someone else talking about him).
说一下自己的想法 – his own ideas (from his own perspective, or emphasized by the narrator).

In most natural contexts, 自己的想法 sounds best because it clearly stresses personal thinking.

Can I omit 问题 and just say 不管多难? Is there any difference?

Yes, you can say:

不管多难，他都想试试说一下自己的想法。

This is also natural and means:

No matter how difficult (it is), he wants to try expressing his ideas.

Differences:

不管问题多难: explicitly mentions 问题 “question/problem”.
不管多难: leaves out the noun; the reference is understood from context.

If the context already makes it clear that you’re talking about questions / problems / tasks, the shorter 不管多难 is very common.

What’s the difference between 问题, 题, and 难题 in this context?

All can relate to “problems/questions”, but with nuances:

问题
- Very general: “question, problem, issue”.
- Can be exam questions, real-life problems, theoretical issues, etc.
- In the sentence, it’s neutral and broad.
题 (short for 题目)
- Usually specific questions/exercises in homework, tests, textbooks, etc.
- Example: 这道题很难。 = This exercise/question is hard.
难题
- Literally “hard problem”; emphasizes difficulty.
- Example: 这是一个难题。 = This is a tough problem.

You could say:

不管题多难，他都想试试说一下自己的想法。
– suggests “no matter how hard the test/exam questions are…”
不管难题多难，他都想……
– a bit redundant (hard-hard), but you might see 不管问题多难 / 不管多难的难题 etc.

In most generic contexts, 问题 is the safest and most natural choice.

Can I change the word order to 他不管问题多难，都想试试……? Is that still correct?

Yes, that word order is also grammatical and natural:

不管问题多难，他都想试试说一下自己的想法。
他不管问题多难，都想试试说一下自己的想法。

Both are fine. Differences are minor:

Starting with 不管问题多难 puts more emphasis on the condition.
Starting with 他 puts slightly more emphasis on the subject (he).

If the focus is on his attitude/character (he is brave, willing to try), the second version might sound a bit more “he-centered”; but in everyday speech, both versions are used.