For (1), regardless of what $U$ is, the structure $(V_n; \in, U \cap V_n)$ cannot be an elementary substructure of $(V_\omega; \in, U \cap V_n)$ because the former satisfies the sentence "there is a largest ordinal" and the latter does not. That is one reason that $\omega$ does not work in place of the class $\infty$ of ordinal numbers (which is more often denoted by $\text{ON}$ or $\text{Ord}$.)
To see why the "weaker reflection principle" in (1) is true, and in fact prove an intermediate version, take any (meta-) natural number $n$. Then one can show that there is an $\alpha$ that is sufficiently "closed" that $V_\alpha$ is a $\Sigma_n$-elementary substructure of $V$—that is, whenever an element of $V_\alpha$ has a $\Sigma_n$ property in $V$, it also has that property in $V_\alpha$.
The proof uses the Tarski-Vaught criterion for elementarity (restricted to $\Sigma_n$ formulas) together with the replacement (or collection) axiom to collect the witnesses into a set. Then we close downward by rank and repeat the process $\omega$ many times to get the required closure point. The reason we must restrict to $\Sigma_n$ formulas for some $n$ is that $\Sigma_n$-definability is definable whereas $\Sigma_\omega$ (first-order) definability is not.
Note that $\Sigma_{n_1}$ elementarity with respect to a class $U$ that is definable by a $\Sigma_{n_2}$ formula will follow from $\Sigma_{n_1+n_2+1}$-elementarity without respect to $U$.
For (2), a hint is that if $\kappa$ is not regular then you can let $U$ be a cofinal function $\alpha \to \kappa$ for some $\alpha < \kappa$.
