O restaurante em que nós vamos jantar fica perto do mercado.

Here em que means in which / where.

The full phrase o restaurante em que nós vamos jantar literally works like:

• the restaurant
• in which
• we are going to have dinner

In natural English, we would usually just say the restaurant where we’re going to have dinner.

Because jantar here is connected to a place with em (jantar em um restaurante = to have dinner in/at a restaurant), that em has to stay when you turn it into a relative clause:

• o restaurante em que...

Yes. Since restaurante is a place, onde is very natural here.

So these are both possible:

• O restaurante em que nós vamos jantar...
• O restaurante onde nós vamos jantar...

Both mean the same thing here.

A third, more formal option is:

• O restaurante no qual nós vamos jantar...

In everyday Brazilian Portuguese, onde is often the most common choice.

Yes, it could be omitted.

Portuguese often leaves out subject pronouns when the verb ending already makes the subject clear. So both are correct:

• em que nós vamos jantar
• em que vamos jantar

Because vamos clearly means we go / we are going, the listener already knows the subject is we.

Including nós can add:

• emphasis
• clarity
• contrast

But in normal speech, omitting it is very common.

This is the very common structure:

• ir + infinitive

So:

• vamos = we go / we are going
• jantar = to have dinner

Together:

• vamos jantar = we are going to have dinner

This is one of the most common ways to talk about the near future in Brazilian Portuguese.

You could also use the simple future:

• jantaremos

But vamos jantar sounds much more natural in everyday conversation.

Here it is a verb in the infinitive.

• jantar = to have dinner / to eat dinner

Examples:

• Vamos jantar às oito. = We’re going to have dinner at eight.
• Quero jantar cedo. = I want to have dinner early.

But jantar can also be a noun:

• O jantar está pronto. = Dinner is ready.

So the word can be both, depending on context.

In Portuguese, ficar is very commonly used to talk about where something is located.

So:

• O restaurante fica perto do mercado. means
• The restaurant is located near the market.

Using ser here would sound unnatural.

A useful pattern to remember:

• Onde fica...? = Where is ...? / Where is ... located?
• A farmácia fica na esquina. = The pharmacy is on the corner.

So fica often means is situated / is located.

Because do is the contraction of:

• de + o = do

So:

• perto de o mercado → perto do mercado

This kind of contraction is extremely common in Portuguese.

Other examples:

• de + a = da
• em + o = no
• em + a = na

So perto do mercado means near the market.

The structure is:

• O restaurante = the restaurant
• em que nós vamos jantar = where we are going to have dinner

So the whole first part is a noun plus a relative clause:

• [O restaurante] [em que nós vamos jantar]

The relative clause gives more information about restaurante.

A similar English pattern would be:

• the restaurant where we’re going to have dinner

So this sentence is basically:

• The restaurant where we’re going to have dinner is near the market.

No. In Portuguese, you normally cannot drop the preposition in this kind of structure.

English allows:

• the restaurant we’re going to eat at
• the restaurant we’re going to eat in
• the restaurant where we’re going to eat

Portuguese needs the connection word with the preposition:

• o restaurante em que vamos jantar
• o restaurante onde vamos jantar
• o restaurante no qual vamos jantar

So a form like:

• o restaurante que vamos jantar is not correct for this meaning.

Not in the same structure.

Here vamos jantar means we are going to have dinner, so jantar is the main action.

If you said para jantar, that would usually mean to/for dinner in a purpose sense, and the sentence structure would change.

Correct:

• O restaurante em que vamos jantar...

Different idea:

• Vamos ao restaurante para jantar. = We’re going to the restaurant to have dinner.

So jantar and para jantar are both possible in Portuguese, but they do different jobs.

It sounds neutral and completely natural.

A few versions by style:

• O restaurante em que nós vamos jantar fica perto do mercado.
  Neutral, clear, slightly careful.

• O restaurante onde vamos jantar fica perto do mercado.
  Very natural and common in everyday Brazilian Portuguese.

• O restaurante no qual vamos jantar fica perto do mercado.
  More formal.

So the original sentence is correct and natural, but many Brazilians would probably choose onde in everyday speech.

A very natural everyday version would be:

• O restaurante onde vamos jantar fica perto do mercado.

This version:

• sounds smooth
• avoids the explicit nós, which is often omitted
• uses onde, which is very common for places

But the original sentence is still perfectly correct and natural. It is just a little more explicit.