It is the nominative marker. Standard Arabic has three cases, nominative (un), accusative (an), and genitive (in). If the first word had a definite article attached to it, those would have become (u/a/i) without the final -n sound. In practice, most native speakers use those endings incorrectly or inconsistently because they are absent in the vernaculars, so you can simply ignore them if it makes it easier for you.
Hi! It actually depends, if you're using Modern Standard Arabic (MSA) -which we call al-fuṣḥá (الفصحى)- then you pronounce the words fully inflected (nunation -in Arabic tanwīn ⟨تَنْوِين⟩- & diacritics -in Arabic tashkil ⟨تَشْكِيل⟩ and ḥarakāt ⟨حَرَكَات⟩-); for example, "الدّجاجُ بارِدٌ" would be pronounced 'al-dajaju baridon' but if you are using a certain dialect in a non-formal way; we usually ignore inflections all together "الدجاج بارد" would be 'al-dajaj barid'. Hope this cleared things up a bit!
Afraid not. Arabic has a class of nouns called the collective, to which you add a simple suffix, usually ة, to form another form called the singulative. When we are talking about chicken in general, not one chicken, not a group of chickens that is small enough to count, but just the concept of chickens or a whole lotta chickens, we say دجاج. But if we are referring to to a particular chicken, it becomes دجاجة (dajaaja[-tun]). A group of chickens that you can actually quantify would be دجاجات, which is the plural. A similar pattern can be observed with many words that refer to animals, plants, and other objects you can find in nature, like rocks. A closely related concept is the instance noun, where adding a suffix to the infinitive gives a word with the meaning "a single instance/occurence of the action indicated by the infinitive," so ضرب would be "hitting," ضربة would be "a hit," for example.