What is the industrial method for extracting caffeine to make decaf? Is this a solvent extraction method? If so what solvent is used and if it is organic, do the manufacturers verify elimination of a certain percentage of residual solvent?
Caffeine is mostly extracted by utilizing very safe processes. Some of them, such as Swiss water, is not even really a chemical process.

Please see the following relevant discussions:


Of the various methods of decaffeinating coffee, which is the safest?
Which decaffeination process changes the flavor of the coffee the least?
How is decaffeinated coffee made?
How are caffeine pills made?


Also, you may want to see this answer to watch an animation of common Swiss water process:


Decaf coffee, how to make our own