There is an extensive literature on Ricardian models with many goods and countries. For a survey which also answers some of the questions you make in your later edit, see:
Eaton, Jonathan, and Samuel Kortum. 2012. "Putting Ricardo to Work." Journal of Economic PerspectivesJournal of Economic Perspectives, 26(2): 65-90. https://www.aeaweb.org/articles.php?doi=10.1257/jep.26.2.65 (freely accessible full-text)
In a Ricardian world there is no unemployment by assumption. What ensures full employment is that wages need not be equal in all countries. The level of wages adjusts in response to productivity differences until a point is reached where each country is the cheapest supplier of some commodities to some destinations (which might include itself). Which commodities these are for each country is determined by the Ricardian principle of comparative advantage.
Original research publications:
Two countries, continuum of goods: Dornbusch, Rudiger, Stanley Fischer, and Paul Anthony Samuelson. "Comparative advantage, trade, and payments in a Ricardian model with a continuum of goods." The American Economic Review (1977): 823-839. http://www.jstor.org/stable/1828066
Many countries, many goods: Eaton, Jonathan, and Samuel Kortum. "Technology, geography, and trade." Econometrica (2002): 1741-1779. http://www.jstor.org/stable/3082019
Ricardian theory has many simplifications: only one input in production, constant returns to scale, perfect competition, full employment, no dynamic effects on technology and resources … These need to be kept in mind when applying its conclusions toWeakening these assumptions may reverse the real worldconclusions that Ricardo reaches and economists have come up with many scenarios where free trade may not be beneficial.
But the papers cited above show that under theseRicardo's assumptions it is possible to give a fully rigorous mathematical account of the theory with no logical problems while preserving many of the qualitative features of the 2-country, 2-goods case.