Grammar is the study of the classes of words, their inflections, as well as their functions and relationships in the sentence. It includes how words and their component parts combine to form a sentence.
In linguistics, grammar is the set of structural rules that governs the composition of clauses, phrases, and words in any given natural language. The term refers also to the study of such rules, and this field includes morphology, syntax, and phonology, often complemented by phonetics, semantics, and pragmatics. Linguists do not normally use the term to refer to orthographical rules, although usage books and style guides that call themselves grammars may also refer to spelling and punctuation.
Yes, in the sentence "The north is known for its culture", the word North should...
Yes, it's standard to capitalize Southern in that usage.