أضع الحقيبة في السيارة قبل أن أذهب إلى المطار.

A fully vowelled version is:

أَضَعُ الحَقِيبَةَ فِي السَّيَّارَةِ قَبْلَ أَنْ أَذْهَبَ إِلَى المَطَارِ.

A careful pronunciation is roughly:

aḍaʿu al-ḥaqībata fī as-sayyārati qabla an adhhaba ilā al-maṭāri

A few useful pronunciation notes:

• السيارة is pronounced as-sayyāra..., not al-sayyāra..., because س is a sun letter, so the ل of ال assimilates.
• المطار keeps the ل sound, because م is a moon letter: al-maṭār.
• In normal speech, especially at the end of phrases, short case vowels are often not pronounced clearly, even though they matter in formal grammar.

Because وضع is a weak verb with و as its first root letter.

In the imperfect/present tense, verbs of this type often drop the initial و. So:

• past: وَضَعَ = he put
• present: يَضَعُ = he puts
• present, 1st person: أَضَعُ = I put

So أضع is the normal correct form.

This happens with other verbs too, for example:

• وَعَدَ → يَعِدُ = he promises

Because the verb itself already shows the subject.

In أضع, the prefix أـ tells you the subject is I. So أنا is not necessary.

• أضع الحقيبة... = I put the bag...

You can add أنا for emphasis or contrast:

• أنا أضع الحقيبة، وليس هو.
  I put the bag, not him.

But in a neutral sentence, leaving it out is more natural.

It can potentially cover more than one of those meanings, because the Arabic imperfect tense is broader than the English present tense.

Depending on context, أضع can mean:

• I put
• I am putting
• I put regularly
• sometimes even I will put

In this sentence, because of before I go to the airport, it often feels like:

• a routine action: I put the bag in the car before I go to the airport
• or a planned near-future action

If you want to make the future more explicit, you could say:

• سأضع الحقيبة في السيارة قبل أن أذهب إلى المطار.
  I will put the bag in the car before I go to the airport.

Because the noun is definite: it means the bag, not a bag.

So:

• حقيبة = a bag
• الحقيبة = the bag

In this sentence, الحقيبة is also the direct object of أضع.

In fully vowelled formal Arabic, it would be:

• الحقيبةَ

with fatḥa because a direct object is normally accusative.

In everyday unvowelled writing, that final case ending is usually not written, so you just see الحقيبة.

Because في means in or inside.

So:

• في السيارة = in the car

This is the most direct way to express physical location inside the car.

Grammatically, after a preposition like في, the following noun is in the genitive case, so in full vocalization it is:

• فِي السَّيَّارَةِ

Because س is one of the sun letters.

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

So:

• السيارة is pronounced as-sayyāra
• not al-sayyāra

But with المطار, the ل is pronounced, because م is a moon letter:

• المطار = al-maṭār

This is a pronunciation rule only; the spelling still uses الـ in both cases.

Because قبل أن is the normal pattern for before + a verb clause.

Here the sentence after before is I go to the airport, which is a full clause with a verb, so Arabic uses:

• قبل أن أذهب = before I go

Compare:

• قبل السفر = before السفر / before travelling
• قبل الذهاب إلى المطار = before going to the airport
• قبل أن أذهب إلى المطار = before I go to the airport

So:

• قبل + noun / verbal noun is possible
• قبل أن + imperfect verb is used when you want a full verbal clause

Yes. After أن, the imperfect verb is normally put in the subjunctive mood.

So the fully vowelled form is:

• أَنْ أَذْهَبَ

not:

• أَذْهَبُ

The important difference is the final vowel:

• indicative: أذهبُ
• subjunctive after أن: أذهبَ

In ordinary Arabic writing, short final vowels are usually not written, so both forms look the same: أذهب.

That is why learners often do not see the difference at first.

Because إلى means to / toward, and the verb ذهب normally takes إلى when you mention the destination.

So:

• أذهب إلى المطار = I go to the airport

This is the standard MSA pattern.

After إلى, the noun is genitive, so in full vocalization:

• إِلَى المَطَارِ

Sometimes, but the meaning changes slightly.

• في السيارة clearly means in the car / inside the car
• بالسيارة often means by car or in the car, depending on context

So if you specifically mean placing the bag physically inside the car, في السيارة is clearer.

For example:

• أذهب إلى المطار بالسيارة.
  I go to the airport by car.

But:

• أضع الحقيبة في السيارة.
  I put the bag in the car.

That is the most natural choice here.

Yes. Arabic word order is fairly flexible, and changing it often changes the emphasis more than the basic meaning.

The original sentence is very natural:

• أضع الحقيبة في السيارة قبل أن أذهب إلى المطار.

You could also say:

• قبل أن أذهب إلى المطار، أضع الحقيبة في السيارة.

This puts more focus on the time relationship: Before I go to the airport...

Both are correct. The original version is straightforward and natural for a learner to study.