How much did the USSR contribute to the Republicans in the Spanish Civil War?

In pure numbers, Daniel Kowalsky has some tables of equipment that you might find interesting. If you scroll down the page, there are a few tables that give various estimates of the arms shipped to Spain. I don't have Beevor's book handy, but the numbers Kowalsky cites line up with my memory of Beevor's estimates. Very roughly the Soviets shipped a few hundred tanks and airplanes (mostly modern types), thousands of machine guns, and millions of rounds of ammunition. Only a relative handful of men were sent as advisors (a few hundred at any one time) and some of them saw combat--especially as tank crews and airplane crews. Compared to the tens of thousands of Italian troops fighting for the Nationalists and the thousands of German troops doing the same, the Russian manpower contingent was quite modest.

The controversial part comes in at this point. First, the USSR was one of a handful of nations (Mexico being another one) that were willing to sell arms to the Republic. France, the UK, and the US were all unwilling to commit to taking a side in the war--a policy given the moniker "non-intervention." Germany, Italy, and the USSR were technically members of an international committee that was to enforce not supplying arms to either side of the Spanish Civil War. All three did anyway, usually under the guise of "volunteer" soldiers and shell corporations that technically sold the arms.

Another bit of controversy surrounds how the Republic paid for the arms. The gold reserves of the Spanish Republic were sent to Russia to finance the arms purchases, and are now known as the "Moscow gold" or "Republican gold." These are propaganda terms that had their genesis in Nationalist attempts to undermine support for the Republic by casting it as a communist entity. There is also some debate as to whether the Republic got its money's worth, but most of the scholars that I have read seem to think that they did.

Two more bits of controversy pop up in regards to communism. The first is that Spanish communists did gain more influence as the war went along, and the fact that the USSR was the main arms supplier for the Republic had a lot to do with that. The result is that the communists had more influence than their mere numbers would justify. They used this as leverage to influence policy, much to the dismay of many socialists, anarchists, and other Republican supporters. The final controversial subject is addressed in your other question.

Can it be considered a proxy war between them and Germany?

You aren't far off here, and that is an insightful question. Joseph Maiolo uses that very term in Cry Havoc. I do think we have to be a little careful, in that there were other factors at play. Italy wanted France to have a troublesome southern neighbor, and they wanted the British to have reduced influence in the Mediterranean--but they weren't ready for open conflict with either nation. Germany wanted France to lose an ally on their southern border, and they wanted to push as far as they could with their burgeoning military without provoking a general European war. France and Britain were much more ideologically compatible with the Republic, but neither wanted to risk full-blown war--particularly against Germany--until their rearmament programs were more up to speed. The Soviets were willing to supply arms to the Republic, but not in overwhelming quantities because that might alarm the west into forming an anti-Soviet coalition rather than fighting amongst themselves. So each of the parties wanted a particular side to win, and some were willing to contribute men and material to that end, but none were willing to have the Spanish Civil War spill over into a general European war. So it was a proxy war of sorts, but a limited one.

As always, followup questions from OP and others are encouraged.

Can it be considered a proxy war between them and Germany?

Actually, the major worry for the USSR at the time was balancing defence of leftist causes and diplomatic relations with France and Britain who's government and populace were set on appeasement. Hence, the USSR was trying their best not to make it seem like a proxy war.

The same could honestly be said for France as well. They too quietly aided the Popular Front; as Luis Araquistáin (the Spanish ambassador to France) wrote, "France did not cease to aid the Spanish Republic, but due to the small amount of material available and [how outdated it was], . . . such aid turned out to be insignificant."

The USSR sighed a the "Anglo-French Non-Intervention Agreement" on August 23 with, according to Payne, what seemed like full intentions of following though. However, once it became clear that neither Germany nor Italy was planning on respecting non-intervention the USSR felt forced to act.

Communism, at this point, was becoming the dominant force in far-left politics. More and more organizations were joining Comintern and non-Bolshevic Marxists were being attacked ideologically from all sides. The USSR was caught between appeasing the left-wing who fully expected them to aid leftist causes with the possibility of provoking the non Communist countries of Europe into forming a coalition against them and losing Spain to pro-Fascist forces, which would disastrously strengthening the anti-soviet block. The USSR, obviously, decided to walk that thin line by using international communism rather than direct, country to country, aid as much as possible.

Walter Krivitsky, an intelligence officer in the Red Army, summed it up well in saying that Stalin's idea was to, "include Spain in the sphere of the Kremlin's influence." "Without public intervention, but by adroit use of his position as the source of military supplies, Stalin believed it possible to create in Spain a regime controlled by him." Payne adds to this saying:

A compromise leftist Republican regime, revolutionary enough to hold domestic support necessary for victory in Spain, but moderate enough to win approval from the western democracies, featuring only one of two official Communist cabinet members, but with Communists controlling the key sources of power, would create the optimum situation for the Soviet Union in southwestern Europe.

So, though 'Proxy War' might be an oversimplification, the Spanish Civil War was definitely part of the larger political and ideological conflict that existed between the two World Wars.

Let me know if you have any other questions, I've been studying the early history of the far-left for almost a year now. Minimized right now is a paper on how the conflicts between Bakunin and Marx impacted the politics of the Spanish Civil War.

Sources:

Payne, Stanley G. The Spanish Revolution. New York: Norton, 1970.

Cattell, David Tredwell. Soviet Diplomacy and the Spanish Civil War. Berkeley: University of California Press, 1957.

Bolloten, Burnett. The Spanish Civil War: Revolution and Counterrevolution. London: The University of California Press, 1991.