يتبل صديقي العجين عند الطاولة.

Because this is a very common Arabic sentence pattern: verb + subject + object.

So:

• يتبل = the verb
• صديقي = the subject
• العجين = the object

Arabic also allows صديقي يتبل العجين عند الطاولة. That version starts with the subject instead, and it is also correct. The version you were given is a normal verbal sentence pattern in Modern Standard Arabic.

يتبل is a present-tense/imperfect verb, meaning something like seasons / is seasoning depending on context.

It comes from the verb تبّل in Form II. In fully vowelled Arabic, it is:

يُتَبِّلُ

A learner may notice that the plain spelling يتبل does not show the short vowels or the shadda. That is normal in everyday Arabic writing.

In normal Arabic writing, short vowels are usually not written.

So:

• يتبل is the ordinary spelling
• يُتَبِّلُ is the fully vowelled spelling

The same thing happens throughout Arabic texts. Native readers usually understand the correct pronunciation from context.

The word صديق means friend.
When you add ـي to the end, it means my.

So:

• صديق = friend
• صديقي = my friend

That final ـي is a possessive suffix.

Because a noun becomes definite in Arabic not only with ال, but also when it is possessed.

So صديقي is already definite because it means my friend. You do not say الصديقي for this meaning.

In other words:

• صديق = a friend / friend
• صديقي = my friend

The possession already makes it specific.

In this sentence, the usual interpretation is:

• after the verb, the first noun is the subject
• the next noun is the direct object

So:

• يتبل صديقي = my friend seasons
• العجين = the dough

In fully vowelled Arabic, case would also help show this:

• العجينَ would be accusative as the object
• صديقي is the subject, so it is nominative in function, even though the possessive ـي hides the usual visible case ending

العجين is the direct object: it is the thing receiving the action of the verb.

So if يتبل means seasons, then العجين is what is being seasoned.

Also notice the ال at the beginning:

• عجين = dough
• العجين = the dough

عند الطاولة is a prepositional phrase of location.

• عند often means at, by, or near
• الطاولة = the table

So the phrase gives the location of the action.

A useful point: عند is not exactly the same as في.

• في = in / inside
• عند = at / by / near

So عند الطاولة is a natural way to say at the table or by the table.

Because of the difference between moon letters and sun letters.

• In العجين, the letter ع is a moon letter, so the l in ال is pronounced:
  al-ʿajīn
• In الطاولة, the letter ط is a sun letter, so the l sound is assimilated and not pronounced separately:
  aṭ-ṭāwilah

So the spelling stays ال, but the pronunciation changes depending on the next letter.

A fully vowelled version would be:

يُتَبِّلُ صديقي العجينَ عندَ الطاولةِ

This shows several things:

• يُتَبِّلُ = present verb
• العجينَ = direct object in the accusative
• عندَ = preposition-like ظرف with a visible fatḥa
• الطاولةِ = after عند, so it is in the genitive

You may notice that صديقي still looks the same. That is because the ـي possessive ending prevents the usual case vowel from appearing clearly at the end.

Yes. Arabic is flexible.

You can also say:

صديقي يتبل العجين عند الطاولة

That starts with the subject instead of the verb. The meaning stays basically the same, though the emphasis can feel slightly different depending on context.

So both patterns are common:

• يتبل صديقي العجين عند الطاولة
• صديقي يتبل العجين عند الطاولة

Yes. The verb agrees with the subject in person, gender, and number.

Since صديقي is understood as:

• third person
• masculine
• singular

the verb is also third-person masculine singular: يتبل.

If the subject changed, the verb form would change too.