Translation:Al-Azhar is a very famous Islamic university.
I don't think so. It sounds a bit messy in English that way. Also, what comes after "is" is the point of identification (predicative) which the sentence is trying to tell. Thus in the sentence above, the identification is for Al-Azhar to be a very famous Islamic university, while in your sentence, Al-Azhar which is an Islamic university is identified as very famous only.
"a smart boy is hungry" is somewhat not directly translated into Arabic, but it depends highly on the context. Maybe we can say something like هناك ولد ذكي وهو جائع (there is a smart boy and he is hungry). This is because Nominal sentences in Arabic (sentences starting with a noun) typically have the subject defined (with AL) and the predicative of the sentence would typically be indefinite. So, starting with an indefinite noun is unlikely. Translating by the context and meaning is the key here (and in every situation mostly between any 2 languages).
"A smart hungry boy" would be ولد جائع ذكي or we can add "and" between adjectives as well: ولد جائع وذكي. Notice that this is not a complete sentence, as the adjectives here are both attributive (describing the noun) and NOT any of them is predicative (telling the status of the noun); i.e. there is no (to be) verb to connect the sentence in English to make a full sentence. Likewise in Arabic, the adjectives follow and mimic the noun in its attributes (gender, number, definition).
No it's a pure Arabic thing, and it's called Nunation or Tanwin (Wikipedia has an article about it). In fact, in dialects nowadays this type of vowels disappeared from the daily speech.
Anyway, you were asking about toon which is actually 2 separate things; It is a T followed by a Tanwin oo or rather u.
Tanwin would come in 3 flavors depending on the grammatical situation: -un, -an, and -in. In the sentence above, the noun جامعة and the adjectives following are all nominative, so they receive the -un to their ends.
You can think of Tanwin as a marker of an indefinite noun (somhow similar to a/an in English). This is an approximation of course and it is helpful to think about it in that way. Tanwin marker would have other jobs in the language other than marking nouns as indefinite.
The second part now related to the T sound you hear. This sound comes from a letter called T-Marbúta ة. This letter in Arabic acts both ways, as (H) and as (T), and it is mostly a marker of a feminine noun. When at the end of the sentence or the word is pronounced alone, this letter typically spelled as (H) - and often would sound just like fatHa or as a short a at the end of the word - However, when the word is in a sentence and the end has to take some vowels because of the grammatical cases and such, this (H) changes to (T). This change is typically required for such word to keep the smooth flow of speech and connect words together. At the end of the sentence, we typically don't find it necessarily to spell the last vowel on the last word in the sentence (usually), so this letter retains its (H) sound; Until it is provoked with a vowel, for one reason or another.