I'm not sure how to approach this mainly because the questions don't only depend on the number of green lights alone, but also where they occur. Since where they occur is uniformly distributed, I'm thinking that the probability for the first question would be $(1-\frac{1}{k-1})^2$,
ie $P$(does not occur at $i$ ) * $P$(does not occur at $j$) but I'm unsure if this is the correct way to do this.
For the 2nd question, I know there is an $exp(\lambda)$ time between jumps (or lights turning green) but once again where the jumps occur also matters. I don't know how to deal with this when solving these. Thanks in advance for any help.
By properties of the Poisson process and uniform distribution, we can say that the lights at every corner turn green at a rate of $\lambda/k$. So the answer to the first question is the probability that by time $t$, the superhero's light hasn't turned green, and the villain's light hasn't turned green, i.e. $(e^{\lambda t/k})^2 = e^{2\lambda t/k}$
For the second, we can instead ask how many switches of the light it takes, on average, until the superhero's light is changed $i$ times more than the villain, or the villain's light is changed $k-i$ times more than the hero.
Keep track of the number of green lights for the superhero relative to the villain after $t$ light changes, call this $N_t$; when the superhero's light turns green, $N$ increases by $1$. When the villain's light turns green, $N$ decreases by $1$. Otherwise, $N$ is unchanged. The first two happen with probability $1/k$, and the third $(k-2)/k$.
You now have a random walk over the states $\{i-k,i-k+1,...,-1,0,1,...,i-1,i\}$, with the transition probabilities
$$ p_{j,j} = \frac{(k-2)}{k} \\ p_{j,j-1} = p_{j,j+1} = \frac{1}{k} \\ $$
and initial condition $N_0 = 0$. Find the expected time to hit either boundary, which is a textbook exercise, then multiply the answer by $1/\lambda$, which is the expected waiting time between every green light.