I can partly answer your questions, but I'm not an expert on the actual Pax Mongolica era.

What cultures had the most success I'm not 100% sure on. As far as I'm aware the Japanese weren't even particularly successful, they just got lucky with two hurricanes destroying the Mongol fleets. I might be oversimplifying though. Otherwise, I know the Mamluks were able to stop them and beat them in battles a few times. Again though, this isn't my period of expertise.

So why am I even responding? Because I do know how you beat a horde later on in time. I've responded to similar-ish posts before here and here. Here's the basic summary of How to Beat a Horde:

There are three main benefits to nomadic warfare. Most importantly, hordes have extreme mobility. They achieve this by being fully cavalry with more than one horse per warrior. Horses actually tire out very quickly if you ride them days at a time, so being able to rotate your horses is extremely useful for traversing large distances. This is very constrained however: If there is no grazing ground, you have to turn back, so hordes are limited to the Arid Zone and a short(ish) distance away from it.

The second big benefit is the numbers. Every nomad is a warrior so the numbers in the "army" are relatively high. This means that there's often a numerical advantage, despite sedentary civilizations having larger populations.

Thirdly, there is the way nomads fight. Horse archers are extremely effective as long as you don't have to take fortified places. Before reliable flintlocks (~1650) it was hard to beat them, provided you could even force them into a pitched battle.

So, there's two ways to deal with this. Way one is to use what they do against them. Eduard Alofs, in his series "Studies on Mounted Warfare" in Warfare in History shows that both the Byzantines and the other sedentary armies near the edge of the Arid Zone in the Middle East used armoured horse archers to deal with the nomads. These were fast enough to be often able to force the nomads into pitched battles and were able to compensate for the lack of numbers with armour. Later on, we can see the Russians using Cossacks to do this, and, I would argue, the Hungarians using Hussars.

The second and ultimately most effective way is to emphasize their restraints. If you can move up agriculture into the steppes, you make their realm of dominance smaller. The Russians did this consciously. More importantly, nomadic tribes do need to trade with sedentary peoples from time to time, to get the things they can't produce themselves, like iron products. If you control all the cities, the nomads have to deal with you. This is, ultimately, how Russia subjugated the steppes.

So, during the Pax Mongolica, the best way to deal with them would be to have enough forts with light cavalry so that they cannot destroy your country without fighting a pitched battle. If they just try to raid you (a horde favourite) you use your light cavalry to stop them from escaping past your forts.

I hope this helps. Apologies if I'm not answering the question you're asking.

On nomadic warfare:

Alofs, Eduard, ‘Studies on Mounted Warfare in Asia I: Continuity and Change in Middle Eastern Warfare, c. CE 550-1350 – What Happened to the Horse Archer?’, War in History 21 4 (2014) 423-444.

Idem, ‘Studies on Mounted Warfare in Asia III: The Iranian Tradition – Cavalry Equipment, Infantry, and Servants, c. CE 550-1350’, War in History 22 2 (2015) 132-154.

Idem, ‘Studies on Mounted Warfare in Asia IV: The Turanian Tradition – The Horse Archers of Inner Asia, c. CE 550-1350’, War in History 22 3 (2015) 274-297.

Sinor, Denis, ‘The Inner Asian Warriors’, Journal of the American Oriental Society 101 2 (1981) 133-144.

Di Cosmo, Nicola, Warfare in Inner Asia. I haven't read this one, yet, but it's a classic.

On Russia subjugating the steppes

Khodarkovsky, Michael, Russia’s Steppe Frontier: The Making of a Colonial Empire, 1500-1800 (Bloomington 2002).

Witzenrath, Christoph, Cossacks and the Russian Empire, 1598-1725: Manipulation, rebellion and expansion into Siberia (London 2007).