My nie mamy zamiaru jeść szybko, dlatego siadamy przy stole i rozmawiamy spokojnie.

Yes. In Polish, the verb ending already shows the subject, so my is often omitted.

• Mamy zamiar... = We intend...
• My mamy zamiar... = still We intend..., but with extra emphasis

Here my can suggest contrast or emphasis, like:

• We are not planning to eat quickly

So the sentence would still be correct without my, but my makes the subject more explicit.

Yes, it is a very common expression.

• mieć zamiar + infinitive = to intend to do something
• nie mieć zamiaru + infinitive = to not intend to do something / to have no intention of doing something

So:

• mamy zamiar jeść = we intend to eat
• nie mamy zamiaru jeść = we do not intend to eat

It is one of the standard ways to express intention in Polish.

Because the sentence is negative.

In the positive form, you normally get:

• mamy zamiar jeść = we intend to eat

In the negative form, Polish very often changes the direct object into the genitive:

• nie mamy zamiaru jeść = we do not intend to eat

So:

• zamiar = accusative/nominative form
• zamiaru = genitive singular

This is a very common pattern after negation in Polish.

Because after mieć zamiar, Polish uses the infinitive of the action being planned.

So the pattern is:

• mam zamiar czytać = I intend to read
• mamy zamiar jeść = we intend to eat
• nie mają zamiaru wyjść = they do not intend to leave

This is similar to English intend to eat, where to eat is also an infinitive.

This is about aspect.

• jeść = imperfective, focusing on the activity/process of eating
• zjeść = perfective, focusing on finishing something

In this sentence, the idea is we are not going to eat quickly as a manner of eating, so jeść fits naturally.

If you said zjeść szybko, it would more strongly suggest finish eating quickly or eat the whole thing quickly.

So:

• jeść szybko = eat quickly
• zjeść szybko = eat up quickly / finish quickly

Dlatego means therefore, that’s why, or so.

It introduces the result of what was said before:

• We do not intend to eat quickly, therefore / that’s why we sit at the table and talk calmly.

It connects cause and result:

• cause: nie mamy zamiaru jeść szybko
• result: siadamy przy stole i rozmawiamy spokojnie

A useful distinction:

• dlatego = therefore / that’s why
• dlatego że = because

Because it separates two clauses.

The sentence has:

1. My nie mamy zamiaru jeść szybko
2. dlatego siadamy przy stole i rozmawiamy spokojnie

Polish punctuation usually puts a comma before this kind of result connector when it links clauses.

So the comma helps show the structure clearly:

• first clause: the reason/background
• second clause: the consequence

Because these verbs mean different things.

• siadać / siąść = to sit down / take a seat
• siedzieć = to be sitting

So:

• siadamy przy stole = we sit down at the table
• siedzimy przy stole = we are sitting at the table

In this sentence, the idea is that instead of eating quickly, we sit down at the table and talk. That makes siadamy the right choice.

Because siadamy is imperfective and works well for a general, descriptive, or habitual idea.

• siadamy can mean we sit down in a general or repeated sense
• usiądziemy usually points to a single future completed action: we will sit down

This sentence sounds like a description of how we behave / what we do, not a one-time future event. That is why siadamy is natural here.

After the preposition przy, Polish normally uses the locative case.

• stół = table
• locative singular: stole

So:

• przy stole = at the table / by the table

This is a very common phrase. It does not mean on the table. For on the table, Polish would use na stole.

They are adverbs.

They describe how the action is done:

• jeść szybko = to eat quickly
• rozmawiać spokojnie = to talk calmly

They come from adjectives:

• szybki = fast → szybko = quickly
• spokojny = calm → spokojnie = calmly

This is very common in Polish.

Because nie normally goes directly before the verb it negates.

Here the negated verb is mamy, so:

• nie mamy zamiaru... = we do not intend...

The negation applies to the whole idea of having the intention.

If you changed where the negation goes, the meaning would change. For example:

• nie mamy zamiaru jeść szybko = we do not intend to eat quickly
• mamy zamiar nie jeść szybko = we intend not to eat quickly

Both are possible, but the first is the more natural phrasing here.

Polish word order is fairly flexible, but the given version is natural and neutral.

Current order:

• My nie mamy zamiaru jeść szybko, dlatego siadamy przy stole i rozmawiamy spokojnie.

You could change the order for emphasis, for example:

• Nie mamy zamiaru jeść szybko, dlatego siadamy przy stole i spokojnie rozmawiamy.

That still sounds natural.

What usually stays important is:

• nie before the verb it negates
• przy followed by locative
• a sensible position for dlatego linking the two clauses

So the sentence structure can move a bit, but not completely freely without changing emphasis or style.