Is my grammar correct and context free?
Your context-free grammar is just fine.  It is not actually necessary for the grammar to have a production with $S$ in its right-hand side.  In fact, a context-free grammar need not have any recursive productions or mutually-recursive productions, though the generated language in that case may be quite limited.  It is also not necessary to use any particular names for your nonterminals (e.g there is no need for $A$ to appear anywhere).