This reply is based on my experiences living and working in Asunción and other parts of Paraguay.

As far as flying in and out of Asunción, your options now are limited. AA used to have a MIA-GRU-ASU flight, but they ended it -- I think it was a year ago that it stopped.

Now, I know that TAM flies to GRU. I think LAN has a flight to SCL, and Gol I think also has flights but I don't know to where or on what schedule.

Unfortunately the complicated and time-consuming routes you mentioned are basically it!

The Sheraton is not far from the airport and it's near a lot of the wealthier areas of Asunción. I don't know anything about the Crowne Plaza.

Good luck!