What is the best way to approach Expected Value calculations when payoffs are highly skewed?

What you actually want is to maximize the growth rate of your bankroll. You can go broke making +EV bets. The Kelly Criterion is the solution you're looking for for something like a lottery – a bet is "rational" iff the Kelly Criterion says you should make it.

It is possible for a lottery to be +EV in dollars and -EV in utility due to the fact of diminishing marginal utility . As you get more of something, the value of gaining another of that thing goes down. The difference between owning 0 homes and owning your first home is substantial, but the difference between owning 99 homes and 100 homes is barely noticeable despite costing just as much money. This is as true of money as it is of everything else since the value of money is in its ability to purchase things (all of which have diminishing marginal utility).

The diminishing value of money is borne out in studies that look for the link between happiness/life satisfaction and income. Additional income almost always improves your life, but the rate of that improvement is approximately at the log scale (i.e. multiplying your income by 10 gives you +1 happiness, regardless of what your income was).

What does this all have to do with a lottery? Well, a lottery gives you a small probability of a massive number of dollars at a fixed cost. Since the hundred millionth dollar is worth much less to you than the first dollar, this can be a bet that has negative expected utility even when you would make money on average.

Even with compute power available, the "value" part of EV calculations gets weird outside of the bounds of normal experience.  Very small probabilities are susceptible to estimation error, and very large impacts are susceptible to value error.

For the lottery specifically, for most non-impoverished people, the ranges of reasonable estimates are such that it's rationally justifiable to play or not play, depending on your enjoyment of the act rather than the monetary EV.  The actual monetary EV is only a guess anyway - given that more tickets get sold when it's large, the chance of splitting the win goes up, and that's ignoring the chance of errors,  cheating, or other shenanigans.  Add to that the fact that you don't know how it changes your life and relationships to win - it's probably quite positive, but it's impossible to predict how much.

It was popular in the 90s among positive-ev gambling folks (card counters and the like) to search the world for +EV lotteries and pool funds to buy LOTS of non-overlapping tickets.  True opportunities were somewhat scarce, and usually the smaller ones, not the giant jackpots.

Most people have non-linear utility of money. Going from 10 thousand to 1 million is less impactful than going from 1 million to 1.99 million, even though it's the same absolute change.

There's a tool called "certain equivalent" which can help with answering "how much do I value money?"

It boils down to repeatedly asking and answering question: "If I had a choice between $X and 50% of $Y at which X would I be ambivalent about which side I pick" for different value of $Y, e.g. for Bob, the 25 year old postdoc it may be

Notice how at higher level certain equivalent is no longer just dividing by two. For Bob utility from having $75k is higher than 50% of utility of having $400k. The reason for this nonlinearity is usually downstream effects of having money. E.g. for Bob $75k would be enough to get a downpayment for house he wants and highly increase chance of him a comfortable life down the line. 50% chance of buying the house outright is not worth the risk for Bob.

(I'm going to nix the cost of the ticket as it's just a constant)

Depends. Do you want to sum the probability weighted payoffs? EV is fine for that. The probability weighting deals with the striking "really, really low" odds (unless you want to further reweight the probabilities themselves by running them through a subjective probability function), and the payoffs are just the payoffs (unless you want to further reweight the payoffs themselves by running them through a subjective utility function). Either or both of these changes may be appropriate to deal with your own subjective views of objective reality, but that's what they are - personal transformations. However, enough people subscribe to such transformations that EU (expected utility, or see cumulative prospect theory) makes sense more widely than just for you. We indeed perceive probabilities differently from their objective meanings and we indeed value payoffs differently from their mere dollar value.

Now, if you just want a number that best represents the payoff structure, we have candidate central tendencies - mean is a good one (that's just EV). But since the payoff distribution is highly skewed, maybe you'd prefer the median. Or the mode. It's a classic problem, but it's finding what represents the objective distribution rather than what summarizes your possible subjective returns.