Good questions! Unfortunately, the answer to the first is actually pretty straightforward, but the second has some interesting dimensions...

There is a lingering impression - born from the disproportionate place that the foreign volunteers have in contemporary media coverage and later the historical memory of the Spanish Civil War - that the International Brigades played a pivotal role in shaping the conflict. Compared to other conflicts, there were indeed a lot of them (maybe 35,000 pro-Republican volunteers all told, about 90% of whom served in the International Brigades). But both sides mobilised hundreds of thousands of volunteers and conscripts, and even in terms of foreign participants, the International Brigades were far outnumbered and outgunned by (mostly) more experienced and better trained troops from Germany, Italy and Spanish Morocco. As such, though I think it's entirely possible to argue that the allure of playing a personal role in defeating European fascism was part of the appeal of going from somewhere like America, this was largely illusory and for some may even have led to significant disillusionment.

The best case you can make for the military significance of the International Brigades is in late 1936, when they arrived just in time to stiffen the defences and resolve at the Siege of Madrid. The Republic desperately needed time to mobilise and train a regular army, and the ad hoc militias that had fought the initial actions had been decimated in a long series of setbacks after the Army of Africa (commanded by General Franco) pushed steadily north after landing on the mainland. A fresh, well-motivated body of troops could indeed make a difference here, not least in demonstrating to the other defenders that the Republic was not alone in the fight - they had allies abroad, and the prospect of further support to come. Indeed, it's rumoured that some locals took these volunteers to be regular Soviet troops - who were indeed in Spain by this point, but limited to technical and advisory roles rather than entire frontline formations.

The problem is, these early volunteers were overwhelmingly not Americans. Americans arrived in significant numbers only in early 1937 - it was after all a longer journey, requiring significant organisation and resources. Americans did participate in the battles around Madrid from February 1937, as Franco's forces sought to cut Madrid's key lines of communication after their direct assault was rebuffed. Yet though battles like Jarama - where the Americans first fought - have gained a mythic status, it's very hard to argue dispassionately that the foreign contingents (either singularly or collectively) tipped the balance. While they continued to fight in significant battles after this point, not only did they represent a smaller and smaller proportion of the Republican forces, even within the International Brigades, Spanish troops slowly became the majority as heavy casualties were replaced by local drafts. Their importance was increasingly political - they were important symbols of international solidarity, and helped ensure that Spain stayed relevant and newsworthy around the world for the duration of the conflict.

None of this is to downplay their individual heroism and sacrifice - the level of casualties they suffered points to the reality that the International Brigades were not there just to pose for photos. But they simply could not make up for the imbalance of expertise, material support and foreign intervention in favour of the Nationalists.

That's not to say that they aren't interesting, and that's where the second part of the question comes in. Their political character is a key part of what made them so unique. They were after all an international army organised by the Communist International as part of the Soviet aid for the Republic – most of its higher leadership were Comintern operatives. As most smaller units were organised by nationality, these were generally led by members of the relevant Communist Party (eg the CPUSA for the American Lincoln Battalion). While the best overarching description of the volunteers is still ‘anti-fascists’, and they included some liberals, socialists and others who were not communists, almost all still tended to agree with the Communist Party about how best to oppose fascism, even if they didn’t subscribe to all of their political programme.

So in that sense, your American volunteer would likely have been a communist or had connections to communists back home, would have been interviewed by a communist to determine their suitability for Spain, would have travelled there alongside mostly likeminded individuals, and would have been trained and commanded by communists once there. There were of course exceptions, or those whose beliefs shifted due to their experiences, but they generally found themselves isolated or even ostracised once in Spain, if their ideas were too significant a departure from the communist line. All this wasn't quite as Orwellian as it sounds - they genuinely did want to get a broad front of leftists to be part of the undertaking, and it was generally Party members who were subject to tight control, surveillance and discipline. But given that the worldview, experiences and training of those running the International Brigades were so profoundly shaped by communism, it was almost inevitable that it seeped into every aspect of their political culture.

However, something I hinted at above matters here - that by mid-1937, more Spaniards than foreigners were joining the International Brigades. While their officers were still likely to be aligned with the Spanish Communist Party, particularly after the XV Brigade (to which the Americans belonged) moved to the Aragon front in late 1937, more and more of their rank-and-file Spanish recruits were anarchists. This necessitated some quite careful diplomacy - the Spaniards were not afraid to complain to their own political organisations if they felt that their views weren't being respected or tolerated. It's worth remembering that despite the popular understanding engendered by Orwell's Homage to Catalonia, the Spanish anarchists and communists were still allies. They cooperated with gritted teeth, they worried about each others' aims, they jockeyed for influence within the Popular Front government, but they were committed to the Popular Front until the very end. Neither side wanted a rupture that would clearly lead to a Franco victory, so both sides tried to iron out these kinds of difficulties as best they could. The same went for encounters with civilians - when moving through pre-dominantly anarchist areas, both military and civil authorities were aware for the potential for misunderstandings, and tried to reduce tensions where they could. The Americans generally did well here - they tended to be generous towards locals, and had a large Cuban contingent who could translate both culturally and linguistically for them.

As such, depending on your volunteer's role and language skills, they might find themselves engaging with Spanish anarchists relatively regularly, and would likely develop some mixture of exasperation and respect for them (some, decades later, would even admit that they sometimes made a good point or two). The answer here would be very different though if you asked about 'Trotskyists' - in the communist worldview, these were simply fascists posing as leftists to sow division from within, and were a legitimate target for repression within the International Brigades and, when communist influence sufficed to make it happen, within Republican Spain more broadly. The anarchists, however, were indispensable to the Republican war effort in a way that the small Trotskyist-aligned organisations weren't, and as such political diplomacy won the day.