Regarding Putin's claim about helping the U.S. Achieve Independence. Perhaps Putin is alluding to Prussia. Prussia was centered around the region of East Prussia, which is now part of modern-day Poland and Russia. General Baron von Steuben was from Prussia. He was born in Madeburg, Prussia, which became part of the Russian Empire in 1795.
Catherine the Great ruled Russia in 1776. And she was German.
The Russian fleet came to New York in September of 1863. The Russians were warmly welcomed by New Yorkers, and seen as a counterweight to the British and British Navy, who were supporting the South outright, at least commercially.
And - History keeps repeating itself - the Russians were occupied in Poland (ie. Belarus and Ukraine) with British-French proxy war aiding a Polish rebellion.
https://www.usni.org/magazines/proceedings/1935/may/visit-russian-squadrons-1863