Find the markov kernel

Question

Given the random process $X_k = V_kX_{k-1} + (1 - V_k)Z_k$, where $V_k$ is a sequence of i.i.d Bernoulli random variables with $P(V_k = 1) = \alpha$, variables $Z_k$ are i.i.d with common distribution $\pi$, $X_0, Z_k, V_k$ are mutually independent for any k.
1) Find the Markov kernel of the process.
2) Show that $\pi$ is reversible w.r.t P
3) Assume that $X_0$ is also distributed according to $\pi$, show that $cov(X_n, X_0) = \alpha^nVar(X_0)$ for any n in N.
Attempt to the solution:
1) Well, I understand that Markov kernel defines a probability of $X_{k+1}$ belonging to some state A, given X_k, i.e. $P(X_{k+1} \in A| \mathscr{F}_k) = N(X_k, A)$. So, we can see that $X_{k}$ can remain at the state $X_k$ with the probability $\alpha$ and become $Z_k$ with the probability $1-\alpha$. How can we write it down using integral representation?
2) This is quite easy, I think, if we now Markov kernel, just check thath tensor product is symmetric.
3) I dont have any ideas. What can I do here?