Question:
- How often does the winning president lose the popular vote?
I know the 2016 election is one example - any other years?
You have to realize Democrats vastly outnumber Republicans in terms of registered voters. It's not exactly surprising Republicans have struggled to win the popular vote. It's a simple game of numbers. However, at the state level where electors decide where the state's vote goes Republicans are often the ones who turn out the most with exception of California, New York, and a handful of other Democrat controlled states (though as of right now things are getting dicey in both Republican and Democrat controlled states).
This seems contradictory until you realize the populations of democrat strongholds such as California and New York have an order of magnitude more people in them. This skews the national popular vote in favor of Democrats. But with, as far as I know, every state implementing winner-takes-all elector voting the states with an even, or more, amount of Republicans swing the election in favor of the Republican candidate. This has happened repeatedly throughout history and is where the Democrat's froth and gnash teeth about "illegitimate" elections and awarding California in particular several more electoral votes due to the population size.
In 2016 Democrats whined about winning the popular vote and losing the election. Several people called for a popular vote style of election but that would never work. Populations are heavily skewed in favor of the coastal Democrat strongholds and so they'd be guaranteed a monarchy with near certainty in that case. It is already difficult to justify voting against the dominant party in your state because in places with heavy one party skew your vote typically won't matter enough to change anything. Having a popular vote style national election would simply scale this to a national level and the southwest, midwest and bible belt may as well not even participate.
One idea might be re-evaluate every state and reissue them electoral votes according to senators. Necessarily this means increasing the number of senators in congress but that might not be such a bad thing considering there's currently no practical way the current batch of senators could ever even fathom of representing even an infinitesimally small amount of their constituents. Though caution must be used here as well - Texas, California, Florida, New York, and Pennsylvania could have disproportionate amounts of influence over congress in this case.