What does it mean to say that events $A_{1}, \ldots, A_{n}$ are (i) pairwise independent, (ii) independent?

Consider pairwise disjoint events $B_{1}, B_{2}, B_{3}$ and $C$, with

$$\mathbb{P}\left(B_{1}\right)=\mathbb{P}\left(B_{2}\right)=\mathbb{P}\left(B_{3}\right)=p \text { and } \mathbb{P}(C)=q, \text { where } 3 p+q \leqslant 1$$

Let $0 \leqslant q \leqslant 1 / 16$. Prove that the events $B_{1} \cup C, B_{2} \cup C$ and $B_{3} \cup C$ are pairwise independent if and only if

$$p=-q+\sqrt{q} .$$

Prove or disprove that there exist $p>0$ and $q>0$ such that these three events are independent.