عندي سؤال عن اسم الطبيب وعنوان المستشفى.

A careful MSA pronunciation is:

ʿindī suʾāl ʿan ismi ṭ-ṭabīb wa ʿunwāni l-mustashfā

A few useful notes:

• عندي = ʿindī
• سؤال = suʾāl
  The ء is a glottal stop, so it is not pronounced like a smooth English suwal.
• الطبيب is pronounced aṭ-ṭabīb, not al-tabīb, because ط is a sun letter, so the l of ال assimilates.
• المستشفى is pronounced al-mustashfā.

In normal reading, many speakers do not pronounce full case endings, especially outside very formal recitation.

Literally, عندي means at me or with me.

It is made of:

• عند = at / with / in the possession of
• ي = my / me

So:

• عندي سؤال literally means At me is a question
• Natural English: I have a question

This is a very common Arabic way to express possession.

Not in the same straightforward way as English have.

In MSA, Arabic usually expresses possession with a phrase such as:

• عندي = I have / with me / at my disposal
• لدي = I have / in my possession

So instead of a normal verb meaning have, Arabic often uses a prepositional or adverbial expression plus a noun.

That is why عندي سؤال is the natural structure for I have a question.

Because the meaning is a question, not the question.

• سؤال = a question
• السؤال = the question

So:

• عندي سؤال = I have a question
• عندي السؤال would mean I have the question, which is a different idea

In this sentence, the speaker is introducing a question, not referring to a specific already-known one.

Because عن means about / regarding / concerning, and that is the normal preposition here.

So:

• سؤال عن... = a question about...

Examples:

• سؤال عن الوقت = a question about the time
• سؤال عن الطبيب = a question about the doctor

In your sentence, عن introduces the topic of the question: the doctor’s name and the hospital’s address.

They are both examples of the iḍāfa construction, often called the construct phrase.

This is how Arabic commonly expresses X of Y:

• اسم الطبيب = the doctor’s name / the name of the doctor
• عنوان المستشفى = the hospital’s address / the address of the hospital

In an iḍāfa:

1. The first noun does not take ال
2. The second noun can be definite
3. The whole phrase becomes definite if the second noun is definite

So:

• اسم طبيب = a doctor’s name
• اسم الطبيب = the doctor’s name

And:

• عنوان مستشفى = a hospital’s address
• عنوان المستشفى = the hospital’s address

Because the two nouns are connected by و and both depend on the same preposition.

So this structure means:

• عن اسم الطبيب
• and عن عنوان المستشفى

Arabic, like English, often uses the preposition once when it applies to both items.

Compare in English:

• I have a question about the doctor’s name and the hospital’s address.

You do not need to repeat about before the second item.

A fully vowelled version would be:

عندي سؤالٌ عن اسمِ الطبيبِ وعنوانِ المستشفى

Here is what is happening:

• سؤالٌ is nominative because it is the delayed subject in عندي سؤال
• اسمِ is genitive because it comes after عن
• الطبيبِ is genitive because it is the second part of an iḍāfa
• عنوانِ is also genitive because it is joined to اسمِ by و
• المستشفى is also genitive in meaning, but the ending is not visibly written the same way because of the final ى

In ordinary printed Arabic, these short vowels are usually omitted.

Yes. Both are correct in MSA.

• عندي سؤال = I have a question
• لدي سؤال = I have a question

A rough guide:

• لدي often sounds a bit more formal or written
• عندي is very common and natural

In your sentence, لدي سؤال عن اسم الطبيب وعنوان المستشفى would also be correct.

Because the word ends in alif maqṣūra: ى.

So the spelling is:

• مستشفى
• not مستشفي

This final ى is pronounced like a long ā sound here:

• mustashfā

This is just the standard spelling of the word hospital in Arabic. It is something learners simply have to get used to recognizing.

Because عندي already contains the idea of me / I.

The ي at the end means my / me, so a separate أنا is not necessary.

• عندي سؤال = I have a question

You could say أنا عندي سؤال, but that adds emphasis, something like:

• As for me, I have a question
• I do have a question

Without emphasis, عندي سؤال is perfectly natural.