cant speak to the employer vs employee taxes, but yes, its true about the foreign earned income tax or whatever its called. I just recently received a pretty nice little federal tax income return based on that. I did work with a tax guy who specifically deals with expats and knows all the ins and outs.
Regarding the residency - Do you already have a DNI that says it expires in 2 years? As far as Im aware, once you have the DNI, its valid until the expiration. However last year, Immigrations did stop into my office one day and asked to see all the foreigners who were registered under the company. I'm not sure exactly why but I guess there is a small chance they could come to the business to confirm that you still work there, though I dont know what the consequences would be. However I do have some friends that had work DNIs and quit, and their DNIs were valid until expiration date for entering and leaving the country.
Once that expires, I dont believe its possible to get any sort of DNI from an American based company that doesnt have operations here. Main reason is that you need to be paying taxes in ARG to get a DNI and thats not possible when an american company is paying you in dollars in the US, nor would the company be registered to sponsor foreigners in ARG.