Snake Featherston
Banned
Superior weaponry in terms of average quality of personal weapons and the like, right? The Red Army will presumably have an advantage in armor and aircraft, even though their other problems may not allow them to use these to their fullest potential. Also, it seems like the Reds would have a numerical advantage...
The Red Army was trounced by Denikin's army using British equipment, which was vastly superior to their own and helped to a degree to overcome the inferiority of Denikin in numerical terms. The Reds had a numerical advantage through the RCW, they were still slapped silly by good generals until interior lines came in. The few times they faced non-Russian forces in that war the result was a disaster for them. It's one thing to beat the 10:1 outnumbered enemies like Kolchak and Yudenich, it's another thing entirely to defeat the BEF.
And finally US army against the Prussians. Their support structure should be about equal, since both are among the pioneers in using railroads in warfare. The real differences are in quantity and quality. The Union fielded in the civil war a total of 2.8 million men, as opposed to the German 1.4 million men (sadly I have no breakdown by state at hand to take Prussia seperatly) had mobilised at the end of the war with France without problems. But while many Union soldiers were volunteers without previous training, every Prussian called up was a trained reservist. And this training included the Auftragstaktik, which one US observer during the war with France viewed as a chaotic behaviour in the Prussian lines instead of a quick reaction by a subordinate officer. Which leads directly to the leadership question. While the Union had some good, even outstanding officers and ousted the really bad ones during the war, they had a small trained officers corps. The Prussians otoh had probably the best trained officer corps in the world and a (for its time) very good staff system. Both together allowed even a sub-average commander to perform adequatly and made someone like Moltke even more efficient. Equipment-wise the Prussians also have a significant advantage. They have breechloading rifles since the early 40s, which means that even reserve units are equipped with it by now. But the true war winner are their breechloading, rifled steel guns introduced since 59 for field and siege.
In a war between only those two additional factors would have to be considered as well, like strategic depth, the nearly untouched Prussian manpower in the cities or the role of the American South, but since such a war is probably asb lets assume they simply meet in a battle: Overall the Prussians have the better chances to win unless they get pinned down in a close quarter fight where only numbers count very early. But somehow I can´t imagine that any half competent Prussian officer willingly gets his army dragged into what would be a gigantic pubbrawl.
The crucial element is who leads the Union army in this timeframe. If we're selecting guys like Rosecrans (at his best), Ord, Canby, Schofield, Grant, Thomas, Sheridan, and so on, then that gives the Union a good chance. If we're putting McClellan, Sigel, Jefferson C. Davis, Buell, Halleck, and the like there, then it's a Prussian massacre of a US armed mob.