From Wikipedia;
In the immediate postwar period it had diplomatic relations with Mexico, Panama, Guatemala, Venezuela, Poland, Czechoslovakia, Hungary, Yugoslavia, Romania, and Albania.[1] By contrast the United States, Great Britain and the Soviet Union did not recognise it.
Did the Government achieve anything during it's exile?
It was stationed in France immediately post-war and then moved to Mexico, so what impact did it have on Franco or Mexican statehood or Culture?
It is worth pointing out that there were technically three governments in exile: the Republic (excepting the Basques and Catalans), the Basques, and the Catalans. I believe you are asking about the first of these, but I can only comment on a tiny sliver of your question--the Basques.
The Basque government in exile established a presence in Paris, including the Basque President José Antonio Aguirre y Lecube. This was in the middle of 1937, after the Basque territory fell to the Nationalists. The main accomplishment of this phase of the Basque government in exile was the facilitation of obtaining visas for Basque refugees. All of this changed in 1940 with the Battle of France. Aguirre was in Belgium at the time with his family. He tried to flee to the French Channel ports, but the German army overtook them before they could reach Dunkirk. This was a desperate situation, as Aguirre was wanted by the Gestapo (Germany and Nationalist Spain weren't allies in the strictest sense, but they cooperated on prisoner exchanges). He took the incredible step of making his way to Berlin (!) and received help from the Panamanian ambassador to get Aguirre and his family to Sweden, then to Brazil--all via an assumed identity. From there he was able to reclaim his identity and join the Basque government in exile in New York City (and became a lecturer at Columbia). There they tried to help exiles and refugees (without great success, the US was not all that enthusiastic about granting these requests) and produced a newsletter to try to get their story more publicity and sympathy.
After WWII the US continued to deal with Franco's regime (particularly as the Cold War began to become entrenched, as Franco's anti-communist credentials were clear even if the legitimacy of his regime was not), and Aguirre eventually returned to France in 1952. Much of the same lobbying work was done, with little in the way of concrete accomplishments. Aguirre died in 1960, and was returned to Spain to be buried.
For Aguirre's biography, Escape via Berlin is excellent--and understandably biased. The Basque History of the World is a pop history book by Mark Kurlansky is highly readable. There are snippets about the fate of the Basque government in exile in the various histories of the Spanish Civil War by Beevor, Thomas, Preston, and Payne.
Sorry I couldn't elaborate more about what I assume is the main thrust of your question!
Also, can anyone tell me why the Soviet Union wouldn't recognize the Republic, considering they contributed to the Spanish Civil War against the Nationalists?