Что, по-вашему, нам делать?

По-вашему means in your opinion, as you see it, or what do you think.

So:

Что, по-вашему, нам делать?
= What, in your opinion, should we do?
= What do you think we should do?

It is a very common parenthetical expression in Russian.

Because this is the standard spelling of this fixed adverb-like expression:

• по-моему = in my opinion
• по-твоему = in your opinion (informal singular)
• по-вашему = in your opinion (formal/plural)
• по-нашему = in our opinion

These forms are conventionally written with a hyphen.

In по-вашему, the word вашему is part of a fixed expression built with по- plus a pronoun form. You do not change it to match anything in the sentence.

For a learner, the easiest way to remember it is as a set phrase:

• по-моему
• по-твоему
• по-вашему

Even though вашему looks like a case form, in this expression the whole unit functions like an adverbial phrase meaning in your view.

Because Russian often uses the pattern кому + infinitive to express what someone should do, has to do, or is supposed to do.

So:

• нам делать literally means something like for us to do
• idiomatically: what should we do?

This is very natural in Russian. Compare:

• Мне что делать? = What should I do?
• Нам куда идти? = Where should we go?
• Тебе не понять. = You can’t understand / It’s not for you to understand.

So нам is in the dative because it marks the person concerned by the action.

Russian commonly uses an infinitive after a question word when asking what action is possible, necessary, or advisable.

So:

• Что нам делать? = What should we do?
• literally closer to: What is there for us to do?

If you used a finite verb, the meaning would be structured differently. The infinitive keeps the sentence more general and idiomatic in this kind of question.

Here что simply means what.

The basic core of the sentence is:

Что нам делать?
= What should we do?

Then по-вашему is inserted to add in your opinion.

So the structure is:

• Что = what
• нам = to us / we
• делать = to do

Because по-вашему is being used as a parenthetical or introductory expression, meaning in your opinion.

Parenthetical expressions are often set off by commas in Russian, just like in English:

• Что, по-вашему, нам делать?
• Это, по-моему, ошибка. = This is, in my opinion, a mistake.

The commas show that по-вашему is not part of the core grammar of что нам делать, but an inserted comment.

Yes. Russian word order is flexible, and the sentence could be rearranged for emphasis.

For example:

• Что, по-вашему, нам делать?
• Что нам, по-вашему, делать?
• По-вашему, что нам делать?

All of these are possible. The differences are mostly about emphasis and rhythm, not basic meaning.

The version you were given is very natural and neutral.

Yes, compared with по-твоему.

• по-твоему = in your opinion when speaking to one person informally
• по-вашему = in your opinion when speaking politely to one person, or to more than one person

So по-вашему matches the polite/plural вы.

By itself, it is generally neutral to polite.

По-вашему can even make the question sound a bit more thoughtful or respectful, because you are explicitly asking for someone’s opinion.

However, tone of voice matters a lot. In the right context, it could also sound impatient or challenging, like:

Well then, what do you think we’re supposed to do?

So the grammar is neutral; the emotional tone depends on context and intonation.

Что нам делать? means simply:

What should we do?

Adding по-вашему adds the idea:

What do you think we should do?

So the difference is:

• Что нам делать? = asking what should be done
• Что, по-вашему, нам делать? = asking specifically for your opinion

Yes, very easily. This pattern is very productive:

• Что нам сказать? = What should we say?
• Куда нам идти? = Where should we go?
• Как нам поступить? = How should we act?
• Когда нам приехать? = When should we come?

And you can also insert по-вашему into these:

• Как, по-вашему, нам поступить?
• Куда, по-вашему, нам идти?

So this sentence is a good model for many similar questions.