I have $P: x > 30$; $Q: x < 20$
Write simply as you can:
a) $P \land Q$
My answer: $x > 30 \;and \; x < 20$
Which is always false:
So if I write: $30 < x < 20$ is it still correct?
b) $\lnot Q \; \land \; P$
My answer: $x \ge 20 \; and \; x > 30$
So: $ 20 \le x > 30$ is it correct?
Another example:
$P: x < 30; \; Q: x > 20$
a) $P \land Q$
My answer: $ x < 30 \; and \; x > 20$
Therefore: $20 < x < 30 $
For the first one, the answer is not well-defined because it depends on what symbols you're allowed to use. As you note, for any real $x$, "$P \land Q$" is always false, and hence equivalent to "$\bot$", which denotes "contradiction".
For the second one, as you noted in a comment, for any real $x$ "$\neg Q \land P$" is equivalent to "$x > 30$", so that's the simplest.
For the third one, yes, "$20 < x < 30$" is the simplest way that we can write it. In general we use such notation to express a chain of equalities or inequalities.