Yes, the noun 'cough' is a common noun; a general word for the act of expelling air through the throat to clear something or as a result of irritation.
The compound noun 'cough syrup' is a count noun:I bought a cough syrup.I bought three different cough syrups.
No, cough is not an adjective. It can be either a noun or a verb.
very common
Yes, cough is a verb. It can be used transitively or intransitively. It can also be used as a noun.
Common
Common noun
The medical term for a cough is tussis.Whooping cough is pertussis; one common brand of cough medicine is Robitussin.
common
Pea is a common noun, and peas is the plural...still a common noun.
A common noun.
Most definitely a common noun.
Camel is a common noun.