سواق التاكسي كان هادي، وانا وصلت بدري.

A common Egyptian Arabic pronunciation would be:

sawwāʔ et-taksi kān hādi, wana weṣelt badri.

A simpler learner-friendly version is:

sawwaa' et-taksi kan hadi, wana woselt badri.

A few quick notes:

• وانا is often pronounced together as wana
• التاكسي is pronounced et-taksi, not al-taksi
• وصلت ends with a clear -t sound: woselt / weṣelt

Yes. سواق means driver in Egyptian Arabic.

It is the everyday Egyptian form. In Modern Standard Arabic, you would more often see سائق.

So:

• سواق = common Egyptian Arabic
• سائق = more formal / MSA

An Egyptian speaker would very naturally say سواق التاكسي for the taxi driver.

Because سواق التاكسي is an iḍāfa-type structure, literally:

driver of the taxi

In this structure:

• the first noun usually does not take الـ
• the whole phrase becomes definite because the second noun is definite

So:

• سواق تاكسي = a taxi driver / taxi driver
• سواق التاكسي = the taxi driver

This is a very common Arabic pattern.

Because ت is a sun letter.

When الـ comes before a sun letter, the l sound is not pronounced, and the next consonant is doubled.

So:

• written: التاكسي
• pronounced: et-taksi

This happens in both Egyptian Arabic and MSA, though the exact vowel in الـ can sound a little different depending on dialect.

كان means was here.

Arabic often uses a noun + adjective sentence with no verb in the present tense, but in the past tense you add كان.

Compare:

• سواق التاكسي هادي = The taxi driver is calm/quiet
• سواق التاكسي كان هادي = The taxi driver was calm/quiet

So كان is marking the sentence as past.

هادي is the normal Egyptian Arabic form of the adjective calm / quiet.

The more formal MSA form is هادئ.

So:

• هادي = Egyptian colloquial
• هادئ = MSA / formal

In everyday Egyptian speech, هادي sounds much more natural.

It can mean both, depending on context.

For a person, هادي can suggest:

• calm
• quiet
• peaceful
• sometimes even gentle

In سواق التاكسي كان هادي, the meaning is something like:

• The taxi driver was calm or
• The taxi driver was quiet

The exact English choice depends on the context you were given.

Yes.

سواق is masculine, so the adjective also appears in the masculine form:

• masculine: هادي
• feminine: هادية

So for a female noun, you might say:

• المدرسة كانت هادية = The teacher was calm
  if المدرسة refers to a female teacher in dialectal use

In your sentence, سواق is masculine, so هادي is the correct form.

وصلت means I arrived.

It is the past tense, first person singular form of وصل = to arrive / to reach.

So:

• وصلت = I arrived
• وصل = he arrived
• وصلتْ = she arrived in careful distinction, though in normal writing the form looks the same

A useful point for learners:
for I arrived, both male and female speakers say وصلت.

Because بدري is an adverb meaning early, and in Egyptian Arabic it very often comes after the verb.

So:

• وصلت بدري = I arrived early

This word order is very natural in Arabic.

You can think of it as:

• وصلت = I arrived
• بدري = early

Yes, بدري is a very common colloquial Egyptian word meaning early.

Examples:

• صحيت بدري = I woke up early
• جينا بدري = We came early
• وصلت بدري = I arrived early

It is much more natural in Egyptian speech than a more formal MSA-style word like مبكرًا.

Because و meaning and is normally written attached to the following word in Arabic.

So:

• و + انا = وانا

In careful spelling, you may also see وأنا, but in informal Egyptian writing people very often write وانا.

It means:

• and I

And in speech it usually sounds like:

• wana

Yes. That would simply mean:

I arrived early

The و adds the meaning and because this sentence is linking two ideas:

• سواق التاكسي كان هادي = The taxi driver was calm
• وانا وصلت بدري = and I arrived early

So without و, it would no longer sound like the second half of a connected sentence.