By definition, for any event probability must be nonnegative. Therefore
\begin{equation*}
P(A \cap B) \ge 0.
\end{equation*}
So,
\begin{equation*}
P(B | A) = \frac{\text{positive or zero}}{\text{positive}}\ge 0.
\end{equation*}
Further,
\begin{equation*}
P (S | A) = P(A \cap S)/P(A) = P(A)/P(A) = 1.
\end{equation*}
For the third part, we will only consider the case when there are two disjoint sets B and C. Then,
\begin{align*}
P(B \cup C | A) & = \frac{P(A \cap (B \cup C)}{P(A)} \\
& = \frac{P( (A \cap B) \cup (A \cap C) )}{P(A)}\\
& = \frac{P(A \cap B)}{P(A)} + \frac{P(A \cap C)}{P(A)}\\
& = P(B | A) + P(C | A).
\end{align*}