Is there a freely usable map of Earth cave systems and/or density?
You might like to check World Cave List which has a pretty extensive list of caves, their depths, and lengths.
For example:

This list has been automatically produced from our World Caves Database.
Total depth and length of all caves currently collected in the database:

Number of caves = 2424
Caves deeper than 300m = 1075
Caves longer than 3kms = 1628
Cumulated depth = 648 932 m
Cumulated length = 17 930 192 m