Σκοπεύω να βρω ένα καφέ που να είναι ήσυχο, ώστε να μιλήσω με τη συνεργάτιδά μου χωρίς φασαρία.

Because Modern Greek normally does not use an infinitive the way English does.
After verbs like σκοπεύω (I intend / I plan), Greek uses να + verb.

So:

• σκοπεύω να βρω = I intend to find
• literally, it is more like I intend that I find

This is a very common Greek pattern.

βρω is the aorist subjunctive form of βρίσκω.

After να, Greek often chooses between:

• aorist subjunctive for a single, complete action
• present subjunctive for an ongoing, repeated, or habitual action

Here, the speaker wants to find one café. That is a single completed action, so να βρω is the natural choice.

• να βρω = to find / to manage to find
• να βρίσκω would suggest something more repeated or ongoing, which does not fit well here

Good question. Here καφέ means café / coffee shop, not the drink coffee.

There are two different words:

• ο καφές = coffee
• το καφέ = café, coffee shop

In this sentence, καφέ is treated as a neuter noun, so you get:

• ένα καφέ
• ήσυχο (also neuter, agreeing with it)

So the grammar is showing that this is not the word for the drink.

Because it agrees with καφέ, which here is neuter.

In Greek, adjectives must agree with the noun they describe in:

• gender
• number
• case

So:

• ένα καφέ = neuter singular
• ήσυχο = neuter singular

That is why you do not see ήσυχος or ήσυχη here.

This is a very common structure in Greek.

• που introduces the relative clause: that / which
• να είναι adds the idea of something wanted, sought, or expected, rather than already known as a fact

So:

• ένα καφέ που να είναι ήσυχο means something like
  a café that is quiet / that would be quiet / that will be quiet enough

The important idea is that the speaker is looking for some café with that quality, not identifying a specific café already known.

Compare:

• ένα καφέ που είναι ήσυχο = a café that is quiet
  more factual, more like you already know such a café exists
• ένα καφέ που να είναι ήσυχο = a café that would be quiet / that is quiet enough
  more natural when talking about what you want to find

Here ώστε να means so that / in order to.

So:

• ώστε να μιλήσω = so that I can talk / in order to talk

It is close in meaning to για να, which is also very common for purpose.

Very roughly:

• για να = the most everyday way to say in order to
• ώστε να = also common, sometimes a bit more formal or polished

In this sentence, ώστε να sounds perfectly natural.

For the same basic reason as να βρω: Greek is using the aorist subjunctive after ώστε να.

• μιλήσω = to speak / to talk, viewed as one complete event
• να μιλάω = to be talking / to talk continuously or habitually

Here the idea is:

• find a quiet café
• then have a conversation

That is naturally seen as one complete event, so μιλήσω fits better.

Because Greek normally does use the definite article with possessives.

So Greek says things like:

• η φίλη μου = my friend
• ο αδερφός σου = your brother
• τη συνεργάτιδά μου = my colleague

This is completely normal Greek grammar.
So even though English says my colleague, Greek usually says the equivalent of the colleague of mine.

Because μου is an enclitic word, and Greek spelling often adds an extra accent before enclitics.

The basic noun is:

• συνεργάτιδα

When you add enclitic μου, it becomes:

• συνεργάτιδά μου

This does not change the meaning or case. It is mainly an accentuation rule of Greek spelling and pronunciation.

You will see the same thing in many similar phrases.

Both exist, but τη is very common before a consonant.

The full form is την, but the final -ν is often dropped in everyday Greek when the next word begins with certain consonants. Since συνεργάτιδα begins with σ, τη συνεργάτιδά μου is perfectly normal.

So:

• τη συνεργάτιδά μου = very normal
• την συνεργάτιδά μου = also possible, depending on style, formality, or regional preference

Because Greek often leaves out the article when speaking about something in a general sense.

Here χωρίς φασαρία means:

• without noise
• without commotion
• without disturbance

It is not talking about a specific noise, so the article is unnecessary.

If you added an article, it would sound more specific:

• χωρίς τη φασαρία = without the noise

But in this sentence, the general meaning is what you want, so χωρίς φασαρία is the natural phrase.