How do I prove regular/non-regular with Nerode-Theorem? How to use it?
You'd have to prove that the equivalence relation for your language is of infinite index to show it isn't regular. If regular, the equivalence classes are exactly the states of tje minimum DFA for the language.

In this case, both are regular (if I understand the notation correctly, $L_1$ is just strings with no $a$; for $L_2$ you can take $u = \epsilon$ and $v$ the given string, so $L_2 = \{1, 2\}^*$). Nice red hering, BTW.