Civilizations like Spain, Austria, Italy, Turkey, Iran, and Sweden had their brief time in the sun, but the Russian civilization has been a consistent mover of the world since the 1500s. Why is that so? What gave them such an advantage over their neighbors that Russian interest were always considered by their rival powers?
I'd hesitate to call Russia alongside other nations you mentioned a "civilization", since it's a country sharing some similarities with its other slavic and nordic neighbours. Depending on your point of view you can either call Russia a part of European civilization with some Asian nomadic influence mixed in or one of the members of Eastern Slavic civilization.
I also wouldn't call Russian state as we know it since 16th century a constant mover who always managed to triumph over its enemies. Russia has had its share of defeats. Livonian war (some also divide it into the Livonian War proper and the War of Livonian succession), the biggest conflict with Russian involvement in 16th century, despite great beginnings (Livonian Confederation de facto ceased to exist, Swedish depredations were checked, Lithuania was dealt a serious blow), ended in a defeat and Ivan IV had to give up all his newly captured possessions. Moscow was burned twice during the 1500s. End of the century saw the mere existance of Russia put to the test as the Time of Troubles, one of the biggest catastrophies in Russian history, started.
Despite all this, Russia still was a very able player in the region. In 15th and 16th centuries it starts to slowly but steadily overpower its neighbours in the form of remnants of the Golden Horde and Luthianian Principality. Reasons for this are threefold: first, Russia was relatively well organized power with very strong influence of the ruler who had great control over his subjects. Siegismund Herberstein who visited Russia in early 16th century wrote in his notes:
Serving Moscowian Emperor is akin to being a slave.
Noting that rules of conduct in Russia were very strict in comparison to more liberal monarch Herbestein grew to be accustomed with.
Second, Russia tried to attack their enemies exactly when they were the weakest in order to multiply their gains and minimize losses. Thirteen Years War with Poland (1654-1667) despite several defeats in the latter phase was ultimately complete success in many ways because more rich and powerful Poland has already been in turmoil because of Cossack uprising. Both victorious wars against Sweden during this period (1590-1595 and 1656-1658) were ended on a high note for Russia due to their proper utilization of resources against weakened enemy, same with their steady, even increasing control over nomadic subjects east and south otf them: tribes of Kalmyks, once brought under Russian jurisdiction, were often used as deterrance against other nomads or sent to the western frontlines. Being somewhat backwards and very thinly populated country in comparison to the Western Europe, Russia simply couldn't allow these resources go to waste.
Third, Russia, despite being very strict religious country, was welcoming to foreigners who changed their faith to Orthodox Christianity or at least proved to be reliable and useful enough for the state's cause. Many mercenaries from Europe, predominantly German and Scottish, served in Russian armies in late 16th and throughout the entirety of the 17th century, and baptized Tatars often managed to gain impressive positions among the ranks of Russian nobility.
So I would say it's a combination of factors. Even then it would be incorrect to present Russia being more powerful than its rivals - for the majority of time it wasn't.