If $A \subseteq B$, is it true that $C \cap A \subseteq C \cap B$? If $A \subseteq B \implies C \cap A \subseteq C \cap B$ ?
Let $x \in C \cap A \implies x \in C$ and $x \in A$ $\implies$ $x \in C$ and $x \in B$ $\implies x \in C \cap B$
is this valid?

Not quite, since your definition of intersection isn't correct.
If $x \in C \cap A$, then $x \in C$ and $x \in A$. If $x \in A$, then $x \in B$, so $x \in C$ and $x \in B$, so $x \in C \cap B$.
