Is "False" in logic analogous to "Null set" in set theory?

You're basically right, but I'll flesh out a pedantic point. Identifying a set $S$ with the unary predicate $\varphi$ for which $\forall e(e\in S\iff\varphi(e))$, a set is $\emptyset$ is identified with the unary $\varphi$ that always returns "false", not with "false" itself. (This is like confusing a constant function with the value it returns; it's a subtle distinction, but an easy one to make if e.g. you define a function as a certain kind of set of ordered pairs.) The usual choice for an explicit statement of this $\varphi$ is that $\varphi(e)$ iff $e\neq e$.

Yes, they are analogous. You can show the analogy a bit more directly like this:

For boolean logic, we have:

$P \land \bot \Leftrightarrow \bot$ (or: $P\cdot 0 = 0$)

and

$P \lor \bot \Leftrightarrow P$ (or: $P+0=P$)

while for sets we have:

$P \cap \emptyset = \emptyset$

and

$P \cup \emptyset = P$

And yes, if you know that all your sets are subsets of some 'universal' set, you have:

$P \cap U = P$

and

$P \cup U = U$

just as in boolean logic you have:

$P \land \top \Leftrightarrow P$ (or: $P\cdot 1 = P$)

and

$P \lor \top \Leftrightarrow \top$ (or: $P+1=1$)