1. Is your grid on integers or centered? that is, e.g., should 118,34 cover the rectangle from 117.5-118.5 and 33.5-34.5 , or should it cover the rectangle from 118-119 and 34-35?

The grid lies is on the degree integers, not centered.

2. There are 20 variables in my data. Unfortunately, they are named X1 through X20. Is there a description of what each variable contains?

Please see the gpw_v4_netcdf_contents_rev11.csv file that was included in the .zip. Rows 62-81 explain the variable definitions.

3. I can guess from the variables' properties that X13, X17, X18, and X20 are years. <https://sedac.ciesin.columbia.edu/data/set/gpw-v4-population-density-adjusted-to-2015-unwpp-country-totals-rev11> states it would go from 2000 to 2020. Alas, my variables do not seem to have years after 2016 in them.

Those are years but read the definitions in the .csv described above. The population density data is in the first 5 variables.

4. Is the population count vs the population density database just different in terms of making the area conversion easier?

Yes, and because the data are in geographic coordinates the area of each pixel is not the same, so we release both a count and density layer.