wide sense stationary of dynamic process
Ok, let's rule out 3. first. Whether that is WSS depends on $U$ and you don't know that. For example, if $U$ has unbounded variance, then so does $X_3$, and bounded variance is a necessary condition for WSS.
For the others:
The aspect of the definition you're looking for is that you can define an autocorrelation function that only depends on the time difference:
$$\phi_{X_kX_k}(n,m)= E\left(X_k(n)X_k^*(m)\right) \overset != E\left(X_k(n)X_k^*(n-h)\right)=:\phi_{X_kX_k}(h)$$
for all times $n$ (and $m$).
Now, let's apply this definition of the autocorrelation function to your $X_1$:
\begin{align}
\phi_{X_1X_1}(n,m) &amp;= E\left(X_1(n)X_1^*(m)\right) \\
&amp;=E\left[\left(0.9X_1(n-1)+V(n)\right)\left(0.9X_1(m-1)+V(m)\right)^*\right]\\
&amp;=E\left[0.81 X_1(n-1)X^*_1(m-1)\right]+E\left[X_1(n-1)V^*(m)\right]+E\left[X^*_1(m-1)V(n)\right]+E\left[V(n)V^*(m)\right]\\
&amp;\text{due to noise being }\color{blue}{\text{independent}}\text{ from signal and } \color{gray}{\text{white}}\\
&amp;=0.81E\left[X_1(n-1)X^*_1(m-1)\right]+\color{blue}{E\left[X_1(n-1)\right]E\left[V^*(m)\right]}+\color{blue}{E\left[X^*_1(m-1)\right]E\left[V(n)\right]}+\color{grey}{0}\\
&amp;\text{white noise must have a }\color{green}{\text{zero mean}}\\
&amp;=0.81E\left[X_1(n-1)X^*_1(m-1)\right]+E\left[X_1(n-1)\right]\color{green}{0}+E\left[X^*_1(m-1)\right]\color{green}{0}\\
&amp;=0.81E\left[X_1(n-1)X^*_1(m-1)\right]\tag Ä\label{fail}
\end{align}
But as seen in \eqref{fail}, $\phi_{X_1X_1}$ depends on the actual points in time $n$ and $m$, and hence, $X_1$ can't be WSS.
Let's address

If $X_1$ was WSS, I believe this should be WSS as well. Please confirm.

Neither $X_1$ nor $X_2$ are WSS, same derivation as above, just that the expectations of noise aren't zero, but still time independent whereas the $X_2$-product isn't.