I don't think you can tie support for Republican Spain to a successful effort to take more part in WWII. Each side was tainted, either by Communism or Facism, and the ideological divide of those two totalitarian systems put everyone off, including those who supported one or the other and those who wanted nothing to do with either.
Mexico taking more of an active role however was doable. Mexico was heavily dependent upon the US for modernizing her military to enable her to do more, and it would take an even larger amount of capital and infrastructure investment to do more, except for utilizing her pool of manpower. It is conceivable the US could have used more of its growing military muscle to train and equip Mexican units for deployment overseas, just a with Brazil. This would have paid dividends later in 1944, when the shortages in infantry hit both the US and Britain. Having Mexico take part in the Italian Campaign, or garrisoning units elsewhere would have enabled some of the US and British units to deploy to France in 1944. Had things gone badly for the Allies after D-Day, it isn't inconceivable Mexican troops could have been brought in, all dependent of course, on how many could be trained and equipped.
I don't see Mexico being able to do much more in the area of her navy. Naval ratings generally take more time to train up, and more investment to do so, than many other types of military units. We were short of small escorts in the early part of the war, and by the time those two problems could be overcome (using US built ships) the U-boat threat had largely been defeated.
How would it affect the nation after the war? Hopefully they have a better infrastructural base and increased industry, and a better economy. There are other problems involved in all of this, covered in this article:
https://www.thoughtco.com/mexican-involvement-in-world-war-two-2136644
As to how it affects US/Mexican relations postwar, a larger involvement, it depends. Were joint operations between the two nations successful? Were casualties equally spread between them? If not, accusations of shirking will be cast, to the detriment of postwar relations. Are Mexican troops involved in some high profile operation with heavy casualties? If they crack the Gothic Line in Italy, and or some of the other German defensive lines in Italy, that can only enhance the prestige of Mexico and the Mexican Militaries, during and postwar.
Regarding the Soviets, I don't see anything positive that can be done with them. Any enlarged relationship between Mexico and the SU will rebound against Mexico with the US, and destroy quite quickly whatever goodwill was developed during the war. The "Cold War" happened very quickly after VE Day, and it would be best to stay on better terms with a next door neighbor than a distant nation.