number of random sets needed to generate subset
The sets $B_i$ can be independent but still have an intersection; just because they are generated independently doesn't prevent them having an element in common (in fact, it ensures that possibility exists).  If you assume that the sets $B_i$ are generated independently, then they are independent, period.  If you don't assume that, you have to define how you assume that they are generated before you can calculate any probabilities.



Regarding your updated question: "But doesn't ... ?"  - Answer: No, it doesn't.  If the r.v.s $B_1,B_2$ are independent, then the events $B_1 \not\subseteq A$, $B_2 \subseteq A$ are independent.

If you want a formal proof, here is one.  If $B_1,B_2$ are independent random variables and $f$ is any function, then $f(B_1),f(B_2)$ are independent random variables.  (You should try to prove it; it is not hard.) Then let $f(X)$ be true if $X \subseteq A$, or false otherwise, and the claimed result follows.