يبدو الطبيب مشغولا.

يبدو is the present-tense form of بدا and here it means seems or appears.

So the sentence is using يبدو to give an impression about the doctor: he seems busy.

Yes, you can.

Arabic often allows both:

• يبدو الطبيب مشغولًا
• الطبيب يبدو مشغولًا

The first is a common verb-first order.
The second puts more focus on the doctor as the topic.

Both are grammatical in Modern Standard Arabic.

It is the noun that يبدو is talking about: the doctor.

In this sentence, it is in the nominative case.
If fully vowelled, it would be:

• الطبيبُ

So the structure is roughly:

• يبدو = seems
• الطبيبُ = the doctor
• مشغولًا = busy

In this pattern, the word that describes the state or quality after يبدو is normally accusative.

So:

• الطبيبُ is nominative
• مشغولًا is accusative

Many learner grammars explain this by treating يبدو here like a كان-type verb: the noun stays nominative, while its predicate/complement is accusative.

That is why you see:

• مشغولًا not
• مشغولٌ

Because مشغولًا here is not an attributive adjective inside a noun phrase. It is a predicate/complement meaning busy.

Compare these two:

• الطبيبُ المشغولُ = the busy doctor
• يبدو الطبيبُ مشغولًا = the doctor seems busy

In the first one, المشغولُ is an adjective directly modifying الطبيبُ, so it matches definiteness.
In the second one, مشغولًا is the predicate/complement, so it is normally indefinite.

In normal Arabic writing, short vowels and tanween are usually omitted.

So the fully vowelled form is:

• مشغولًا

But in ordinary writing you will often see:

• مشغولا

The final ا is part of the usual spelling of many words with fatḥatayn in the accusative.

So:

• fully marked: مشغولًا
• normally written: مشغولا

A careful full pronunciation is:

• yabdū aṭ-ṭabību mashghūlan

A few helpful points:

• يبدو = yabdū
• الطبيب is pronounced aṭ-ṭabīb, not al-ṭabīb, because ط is a sun letter, so the ل of ال assimilates.
• مشغولًا = mashghūlan

If you pause at the end of the sentence, the final -n of tanween is often dropped in actual recitation.

Yes. It agrees in gender and number.

Here both are masculine singular, so we get:

• الطبيب ... مشغولًا

If the noun changes, مشغول changes too. For example:

• تبدو الطبيبة مشغولةً = the female doctor seems busy
• يبدو الطبيبان مشغولَين = the two doctors seem busy
• يبدو الأطباء مشغولين = the doctors seem busy

It does not have to match definiteness here, because it is predicative, not attributive.

مشغول is originally a passive participle from the root ش غ ل, with a basic sense like occupied.

In many contexts, though, it functions just like an ordinary adjective meaning busy.

So in this sentence, you can think of it simply as busy.

No.

يبدو is already the main verb, and it means seems. Arabic does not add another present-tense to be verb here.

So:

• يبدو الطبيب مشغولًا

already means The doctor seems busy.

A related alternative is:

• يبدو أن الطبيب مشغولٌ

Literally: It seems that the doctor is busy.

That version is also correct and common.