Proving that picking odd-numbered symbols can create the universal set from a non-regular language
You want to show that $\operatorname{odd}(L_1) = \{a,b\}^*$. This is a set equality claim, and you show it the way you can approach any such claim:


Show $\operatorname{odd}(L_1) \subseteq \{a,b\}^*$.

That's trivial.
Show $\operatorname{odd}(L_1) \supseteq \{a,b\}^*$.

Let $w \in \{a,b\}^*$. Can you construct $w' \in L_1$ so that $w = \operatorname{odd}(w')$?


   You only need to fix symbol counts, so brute force works:
 $w' = w_1 \overline{w_1}  \cdots w_n \overline{w_n}$ with
 $\overline{x}$ the "inverse", i.e. it maps $a$ to $b$ and vice versa.


Thus, $w \in \operatorname{odd}(L_1)$. Since $w$ was chosen arbitrarily, 
the claim follows.


A similar proof works for $\operatorname{even}(L_1)$.