نظفي الطاولة بعد الغداء، ثم اجلسي معنا إذا كنت غير مشغولة.

It is part of the verb.

Here, نظفي and اجلسي are commands addressed to one female. In Modern Standard Arabic, the imperative for a singular feminine addressee often ends in -ي:

• نظِّفْ = clean! (to one male)
• نظِّفي = clean! (to one female)
• اجلسْ = sit! (to one male)
• اجلسي = sit! (to one female)

So the sentence is speaking to a woman or girl. The -ي here is not the pronoun my/me.

Yes. In full pronunciation, it is نظِّفي.

The verb comes from نظَّفَ meaning to clean. That verb has a doubled middle consonant, shown by a shadda on ظ. In carefully marked writing, the imperative here would be:

• نظِّفي

But in normal everyday Arabic writing, short vowels and shadda are often omitted, so you may see:

• نظفي

Even without the mark, the intended pronunciation is still with the doubled consonant.

That initial ا is part of the imperative form. It is often called hamzat al-wasl.

The basic verb is جلس = to sit. In Form I verbs like this, the imperative often needs a helping vowel at the beginning, because Arabic does not like starting the word with certain consonant clusters.

So you get:

• يجلس = he sits
• اجلسْ = sit! (to one male)
• اجلسي = sit! (to one female)

So the opening ا is not a separate word; it is part of how this imperative is formed.

Because normal Arabic spelling usually leaves off short vowels at the ends of words.

So كنت can stand for different pronunciations depending on context, such as:

• كنتَ = you were (masculine)
• كنتِ = you were (feminine)
• كنتُ = I was

In this sentence, the context shows that it must be كنتِ:

• the commands نظفي and اجلسي are feminine
• مشغولة is also feminine

So the full reading is إذا كنتِ غير مشغولة = if you are not busy.

غير is a very common word meaning non-, not, or other than, depending on context.

Here:

• مشغولة = busy / occupied (feminine)
• غير مشغولة = not busy

The adjective is feminine because it describes the person being addressed, who is a single female.

Compare:

• مشغول = busy (masculine)
• مشغولة = busy (feminine)

So:

• إذا كنتِ غير مشغولة = if you are not busy

معنا means with us.

It is made of:

• مع = with
• نا = us / we

So literally:

• مع + نا = معنا = with us

This is a very common pattern in Arabic: a preposition plus an attached pronoun.

For example:

• معي = with me
• معك = with you
• معنا = with us

They both help show sequence, but they do different jobs.

• بعد الغداء = after lunch
• ثم = then

So the sentence is structured like this:

• نظفي الطاولة بعد الغداء = Clean the table after lunch
• ثم اجلسي معنا = then sit with us

A useful detail:

• بعد is a preposition, so الغداء is grammatically after a preposition
• in fully vocalized Arabic, it would be بعدَ الغداءِ

Also, ثم usually suggests sequence with a sense of next / afterwards, often a little more separate than simple و.

Yes, الطاولة is the direct object of نظفي.

The verb نظِّفي means clean!, and the thing being cleaned is الطاولة = the table.

So:

• نظفي الطاولة = Clean the table

In fully vocalized Arabic, the direct object would be marked with fatḥa:

• الطاولةَ

But in normal writing, that ending is usually not shown, so you simply see الطاولة.

إذا means if and introduces a condition.

Here:

• إذا كنتِ غير مشغولة = if you are not busy

Yes, the condition could also come first. For example:

• إذا كنتِ غير مشغولة، فاجلسي معنا.

That means the same basic thing: If you are not busy, then sit with us.

In your sentence, the condition comes after the command:

• ثم اجلسي معنا إذا كنت غير مشغولة

That is also perfectly natural, and it sounds like: then sit with us if you are not busy.