1)First one is regular
the Language is ={0000,00100,00000,000100,…..}
We can write the regular expression =$00(1+0)^{*}00$ as $u\in \Sigma ^{*}$ which can eat up all the strings in between . The language contain all the string which start and end with $00$.
2) It is not regular language.
proof by myhill – nerode theorem:-
$L=\left \{ 0^{k}1u0^{k} |k>1\wedge u\in \sum ^{*}\right \}$
Consider the infinite set of strings
$S=\left \{ 0^{k}1 |k\geq 1 \right \}$
Claim $S$ is distinguishing set for L.
Take any pair of string $\left ( 0^{k}1,0^{j}1 \right )$ of strings where $j\neq k$.
Let $z=0^{k}$
Then $z=0^{k}10^{k}$ is in L but $z=0^{j}10^{k}$ is not in L.
So all pairs of strings in S are distinguishable to L.
Hence there are infinitely many equivalence of $\equiv _{L}$, so L is not regular .
https://people.csail.mit.edu/rrw/6.045-2020/lec6-color.pdf