1. Do they really have concepts of capitol city? Which "Pokrovka" of the real life did Firaxis uses?
The Pokrovka they used is simply a modern (Russian) name for a Scythian archeological site. Linguistically, it is impossible that there would be a Scythian town, camp or city with that name. The Scythian 'towns' we do know about are all in Herodotus: Portmei, Roxanaki, Gelonus, and the fortresses of Napit, Palakion and Habei in the Crimea, and Gilea, a religious site that may have not had any actual settlement associated with it.
2. Were they the ancestors of modern Ukranians as well as Kazhakstanian?
Kazakhistani, yes sort of Scythian, somewhat removed by many subsequent overlays of Turkic-speakers who had virtually the same lifestyle and mounted pastoral 'technology' and culture, but modern Ukrainians are genetically Slavs who came down from the forested north, with a dab of Scandinavian ("Rus") mixed in and culturally a large dose of Byzantine Greek.
3. Meanings of stag head emblem that is a racial symbol in this game
Scythian art was heavy with semi-stylized animal motifs. The stag and horse were prominent among them.
4. Were mythical Amazons of the Greek mythology actually Scythians? Were 'Thermiscyra' as presented in both Greek mythology and DC actually a Scythian city?
The Amazon story comes from Herodotus, who lived among the Scythians and Greek colonies bordering 'Scythia' in what is now the Ukraine and Crimea. Everybody thought he was simply spinning a tale until Soviet archeologists unearthed a burial of a Scythian woman who had been buried with over 20 spear or arrow points, armor, horses, saddle, and all the trappings of a mounted warrior. Specifically, Herodotus says that among the Roxolani, a Scythian tribe, the men fought as armored lancers but the women fought as mounted archers, and that the women were not allowed to marry until they proved that they had killed an enemy in battle. Consequently, nobody wanted to fight the Roxolani, because the women would shoot you full of arrows from a safe distance, then close in, cut off a prominent part of your anatomy like your head or your genitals, and take them back as proof of a kill. Nothing sportsmanlike about them at all.
5. Did they shown up in what's now China as well? Since they're ancient race, What is the name the (more civilized) Han Chinese call them? (And logograms the Chinese wrote to represent Scythians) Did they still exists in the days of Qin Shi Huangdi? Were The Great Wall constructed to deal with them too?
The basic horse nomad or pastoral culture and equipment show up from about 800 BCE to 1500 CE in a swath from Mongolia to the southern Ukraine. Everybody on the Eurasian steppe rode horses, herded cattle, hunted and fought with bows and spears. So it didn't matter if they called themselves Cimmerians, Scythians, Hsung-Nu, Huns, Pechenegs, Patzinaks, Turks, Bulgars, Khazars, Uighurs or whatever, they all rode rings around 'civilized' armies and shot them to pieces. On the receiving end of the arrows, they all looked alike and acted alike. Chinese Dynasties directly interacted with, and occasionally married, Hsung-nu, Turks and Uighurs.
6. If Cossacks originated from runaway serves and slaves of various empires around steppes where Europe and Asia meets and not a name of any tribe (There were Russians, Turks, and maybe other Crimean peoples). Were cossacks also originated from Scythians too?
Only distantly. By the time the first Cossack 'Hosts' began to form at Zaporozhe along the lower Dnepr or along the lower Don Rivers, the Scythians had been gone for most 1500 years. But, as stated above, everybody who came after the Scythians pretty much acted the same way on the same terrain, so culturally and technologically the Cossacks were 'descendants' of the Scythians.
Parenthetically, the Civ V and Civ VI depiction of Cossacks is really, really bad. The games show them as 19th century light cavalry, by which time they were practically indistinguishable from every other light cavalry in Europe - better horsemen than most, but armed and organized the same way and used the same way in combat. In their prime in the 16th and 17th centuries, the Cossacks were notorious as light Lancers who could outrun anybody they couldn't cut up and, by reputation that they made the most of, the best pillagers to ever ride a horse. The only thing the games get right is their effectiveness at massacring the wounded and stragglers.