Assuming I was looking for arrival opportunities from 2020 to 2030, how can I calculate the possible arrival dates to Mercury given a departure date from a gravity assist from Venus?
You don't start with a departure date--that's one of the answers, not one of the inputs.
I don't know a formula for calculating a gravity assist orbit so I will use the easy case and omit Venus, as well as assuming the planets are in circular, coplanar orbits:
Find the orbital period of the transfer orbit. For our simple case the orbital radius is (mercury orbit + earth orbit)/2 and Kepler will give you the orbital period from that. We will go halfway around this orbit. You look at the two orbits and hunt for a time where the planet you are launching from is opposite from where the target will be 1/2 of the transfer orbit period in the future. I believe there is an algebraic solution to this but I don't recall it.
While this gives a single point in time as the answer in practice it's a bit fuzzy, you can deviate a bit from this without a big fuel cost, but after that the fuel cost goes up prohibitively, launches simply aren't done.
For any given pair of planets this happens at fixed intervals, once you know one time and the repeat period you can very easy calculate additional times.
(Note that this is why there was such a rush to get Perseverance launched.)
Once you start adding gravity assists the problem becomes much, much harder as you need suitable windows with both pairs of planets and the correct speed and angle from the encounter. I believe this is simply brute-forced, no algebraic solution exists.