Learn Hardcore Chinese: nǐ yǒu wèntí ma?

Yes. 你有没有问题？ (nǐ yǒu méi yǒu wèntí?) is common and slightly more probing/emphatic, like “Do you have a question or not?” For addressing a group, 有没有问题？ is very natural.

How should I pronounce the whole sentence with tone sandhi?

你 (3rd) + 有 (3rd) → say it as ní yǒu (2nd + 3rd) due to 3–3 sandhi.
问题 is wèn tí (4th + 2nd).
吗 is neutral tone.
Overall: ní yǒu wèn tí ma? with a generally falling ending.

Does 问题 mean “question” or “problem”? How do I avoid confusion?

It means both. Context decides.

Classroom: 有问题吗？ ≈ “Any questions?”
Troubleshooting: 有问题吗？ ≈ “Is there a problem/Is something wrong?”
Avoid saying 你对我有问题吗？ unless you mean “Do you have a problem with me?” If you mean “Do you have a question for me?”, say 你有要问我的问题吗？ or 你有什么想问我的吗？

What’s a more inviting way to ask for questions from a group?

有什么问题吗？ (More open, invites content)
你们有什么问题？ (Invites actual questions, not just yes/no)
Polite: 请问，大家有什么问题？

Is a measure word needed before 问题?

Not in the generic yes–no question. Use a measure word when specifying number:

一个问题 (one question), 几个问题 (a few questions)
“I have a question”: 我有一个问题。
“Do you have one specific question?” (unusual to ask): 你有一个问题吗？

How do I negate or say “no problem” correctly—不 or 没?

With 有, negate using 没(有), not 不.

Correct: 没有(问题)。
Incorrect: 不有(问题)。
“No problem” is 没问题 (very common) or 没有问题 (more formal/literal).

What’s the difference between 没问题 and 没有问题?

没问题: colloquial “No problem/It’s fine/Can do.”
没有问题: literal “There is no problem,” often more formal or emphatic. In replies, both are acceptable depending on tone and setting.

How do I make it more polite or address someone formally?

To one person politely: 您有问题吗？
To a group politely: 各位有问题吗？ / 诸位有问题吗？
Soft/inviting: 请问，您有什么问题？

Can I use 呢 instead of 吗?

Not for yes–no questions. 呢 does not replace 吗. Use 吗 or the A‑not‑A pattern. 呢 is for topics/continuation, e.g., 你呢？ (“and you?”).

Can I topicalize it as 问题你有吗？

That word order is marked/unnatural in this context. Stick to Subject + 有 + Object + 吗: 你有问题吗？

Is 有 here “have” or “there is/are”? Can it be existential?

Both meanings exist for 有.

Possessive: 你有问题吗？ (Do you have...)
Existential/location: 这里有问题吗？ (Is there a problem here?)
For addressing an audience, 有问题吗？ is like “Any questions?” (existential).

Do I still need a question mark if I already have 吗?

Yes. Use ？ at the end in writing: 你有问题吗？ (Full-width Chinese question mark in Chinese text.)

How do I specify the kind of question?

Add a modifier after 问题 or use 关于/在…方面:

你有技术方面的问题吗？ (any technical questions?)
你对作业有疑问吗？ (any doubts about the homework?)
Note: 对+N+有问题 often means “there’s something wrong with N,” not “a question about N.” Prefer 对…有疑问 or 关于…有问题/疑问 for “questions about …”

What changes if I add 了: 你有问题了吗？

了 adds a change‑of‑state meaning: “Do you have a problem now (has one arisen)?” It’s not the neutral “Do you have any questions?” Use 你有问题吗？ for the neutral version.

Are there colloquial variants I might hear?

Yes:

有没有问题？ (very common)
有问题没？ (informal/regional)
With a tag: 有问题吧？ (you probably have a problem, right?) or 有问题啊？ (surprised/concerned) — these change the nuance, not purely neutral questions.

What are natural follow‑ups after someone says 有?

什么问题？ (What’s the question?)
请说/你问吧。 (Go ahead.)
具体是什么问题？ (Specifically, what is it?)

What’s the Traditional Chinese version?

你有問題嗎？ (nǐ yǒu wèntí ma?)