About the time differences in the Bellman equation
During the derivation of the Bellman Equations, when the expected cumulative rewards are calculated on an infinite horizon, meaning $t \to \infty$, shouldn't they be equal for different time steps; like $V^{\pi}(s_i=s) = V^{\pi}(s_{i+\Delta i}=s), \forall s$?


Yes, provided the state $s$ has the Markov property, which means it contains all relevant data about immediate state progression and reward, then this is true. I would read your equations as over-specified in fact, and often see the equations written differently:

$$v_{\pi}(s) = \mathbb{E}[\sum_{k=0}^{\infty} \gamma^k R_{t+k+1} | S_t = s, A_t \sim \pi]$$

although there are several roughly equivalent ways to describe value functions in RL (for instance, and trivially, you could write $\mathbb{E}_{\pi}$ for the expectation which already implies the condition $A_t \sim \pi$). The one in this answer makes it clear that the value of the state $s$ is independent of the time step on which it is measured.

It is worth noting that although the expectation is always the same and independent of $t$, that actual returns seen can be different, and if you know from some other context that you are seeing the first, or last visit to a state - or from the time step whether this is more or less likely given $\pi$ - then that may change the expectation when loops are possible.