Is $e^e$ irrational?
About (1), it is still unknown whether $e^e$ is irrational or not, according to Wikipedia.
https://en.wikipedia.org/wiki/Irrational_number#Open_questions
Even more interesting, according to Gelfond's Theorem, $a^b$ is transcendental (therefore irrational) if $a$ is algebraic (and $\not\in\{0,1\}$) and if $b$ is irrational and algebraic.
http://mathworld.wolfram.com/GelfondsTheorem.html
This theorem can be used to prove that $e^\pi$ is transcendental and therefore irrational.