I can answer the Mexico portion. Mexico owed major debt to the European powers and were holding off payments. Napoleon III created a coalition of powers to capture Veracruz, the port city that leads to Mexico City. Soon it was realized that the debt was not the main objective for Napoleon. He wanted to create a 2nd French Colonial Empire in the Americas. He "invited" the archduke to mend relations with Austria following the Second Italian War of Independence, in which Napoleon back Sardinia and Austria was forced to cede Lombardy. Maximilian's claim to the Mexican Crown came in the form of having been part of the Hapsburg family, whom were once the kings of Spain back when Mexico was part of the Spanish Rule. He also adopted the grandsons of the first Mexican Emperor, Agustin I. Considering Maximilian was supported by France, Austria, and Belgium (his wife was Charlotte of Belgium) his rule would be stable compared to if someone of Mexican nobility were to seize the crown.