يجلس صديقي على كرسي الأسنان.

It can mean either, depending on context.

• يجلس is the imperfect verb form.
• In Modern Standard Arabic, that form can express:
  • a present ongoing action: is sitting
  • a general/habitual action: sits

So this sentence could mean either My friend is sitting on the dental chair or My friend sits on the dental chair, depending on the situation.

Because verb–subject–object/complement order is very common in Arabic.

So:

• يجلس صديقي... = a normal verb-first sentence
• صديقي يجلس... = also correct, but it puts more focus on my friend

Both are grammatical. The version with the verb first often sounds more neutral in MSA narrative style.

Arabic verbs already include information about the subject.

يجلس tells you that the subject is:

• third person
• masculine
• singular

So by itself, يجلس means he sits / he is sitting.

Then صديقي is added to tell you exactly who that he is: my friend.

صديقي is made of:

• صديق = friend
• -ي = my

So صديقي literally means my friend.

This -ي is an attached possessive pronoun. Arabic often shows possession this way instead of using a separate word for my.

Because a noun with a possessive suffix is already definite.

So:

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

Since my friend is already specific, you do not add ال.
So الصديقي would be wrong here.

Because Arabic commonly uses على with chairs, seats, beds, and similar things.

• على basically means on
• but in natural English, the best translation may sometimes be in, especially with a chair used for treatment

So Arabic says على كرسي الأسنان, even if English might prefer in the dentist’s chair in some contexts.

This is an iḍāfa construction, often called the construct phrase.

It works like this:

• كرسي = chair
• الأسنان = the teeth

Together, كرسي الأسنان means something like:

• the dental chair
• the chair for teeth
• by extension, the dentist’s chair in context

In an iḍāfa:

• the first noun usually does not take ال
• the second noun can take ال
• the whole phrase gets its definiteness from the second noun

So كرسي الأسنان is correct, while الكرسي الأسنان is not the normal structure.

Yes. الأسنان literally means the teeth.

Arabic often uses a literal noun like this where English might use an adjective such as dental. So:

• كرسي الأسنان literally = the chair of the teeth
• natural meaning = the dental chair

If you want a more explicit phrase for the dentist’s chair, Arabic could also say كرسي طبيب الأسنان.

A careful MSA pronunciation is:

yajlisu ṣadīqī ʿalā kursiyyi l-asnān(i)

A few notes:

• يجلس = yajlisu
• صديقي = ṣadīqī
• على = ʿalā
• كرسي in full MSA is pronounced roughly kursiyy
• الأسنان = al-asnān

In pause, speakers often reduce or drop some final case vowels, especially outside very formal recitation.

Yes, a few.

• ص in صديقي: an emphatic s. It is heavier/deeper than plain س.
• ع in على: a sound English does not have. It comes from the throat.
• ق in صديق: a deep q sound, farther back than English k.
• سنان in الأسنان has a long ā in -snān.

These sounds usually take practice, especially ع and ق.

Yes, in full formal MSA there are.

A fully vocalized version is:

يَجلِسُ صَديقي عَلَى كُرْسِيِّ الأَسْنَانِ

Here is what is happening:

• يَجلِسُ ends with -u in the normal indicative form
• صديقي is the subject, so it is nominative, though the case is not obvious in the usual spelling because of the attached -ي
• على makes the following noun genitive
• so كرسي becomes كرسيِّ
• الأسنانِ is also genitive because it is the second part of the iḍāfa

In everyday spoken use, learners often do not pronounce all these endings, but they matter in careful MSA grammar.