An analysis of the state and federal tax contributions from undocumented immigrants across the US.
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Are you sure that you proved your original statements? You stated:
Yet, as far as I can see, you only proved that undocumented immigrants pay some taxes:
Just because some of them pay some taxes doesn't prove anything in your first paragraph. It is still quite possible that they take away jobs from working class, especially lower- and middle-class Americans who need that work. It's possible that they do this by BOTH increasing the labor supply which decreases wages, and by undercutting wages by working for cash and not paying taxes. And by engaging in these activities it is quite possible that they cost the US government billions of dollars.
If those jobs were filled completely by Americans who were paying taxes, not only would tax collections be among a larger percentage of the population, but the higher wages resulting for the same jobs filled by American citizens would result in higher revenues collected for the same jobs that had been filled by 'undocumented' immigrants (you don't call them 'illegal', yet you admit that some of them are using 'fake' SSNs which is prima facia illegal).
I don't see anything particularly compelling here as far as arguments for any tax code reforms. And what is notably missing is any methodological discussion of the three points in your very first sentence.