NATO has been setting itself up primarily as an instrument of the American Empire, and as an anti-Russia alliance. Of course Putin is going to want to dismantle it. The invasion of the Baltics to destabilize NATO is not quite as farcical as NATO using the Baltics as a staging ground to invade Russia, but it's pretty close.
"Instrument of the American Empire" is incredibly vague. NATO provides a lot more to the Europeans than it does to the Americans, whom are very capable of launching interventions and invasions without their help.
The most prominent NATO operations have been in the former Yugoslavia, with extensive United Nations backing (until 1999), Afghanistan, with moderate United Nations backing, Anti-piracy duties near Somalia with UN backing, and Libya, with moderate United Nations backing. The Libyan campaign was spearheaded not by America, but by France and Britain. America was willing to help (and needed desperately as it turned out when the poor stockpiles maintained by other NATO forces were exhausted), but it was not really "their" operation, in the same way that Afghanistan was. NATO is not directly equivalent to American foreign policy.
I would describe NATO's foreign commitments as an instrument of western consensus, it acts when the interests of many of its members align. When a consensus cannot be reached, countries tend to act individually instead of through the NATO framework.
Bad actor as in a player that's responsible for ratcheting up tensions. Yeah, I can see Russia getting the brunt of that for their military adventures since 2014 in Crimea and Ukraine, but the bad blood has been getting worse ever since about 2008, and NATO was probably more at fault back then.
The only notable flashpoint in 2008 was the Russian action in South Ossetia / Georgia, which pissed off several NATO member states. Still, even after this it was not inevitable that relations permanently decline. Circa 2011-2012 things were pretty stable and even looked to be improving somewhat. 2014 is where things tank irrevocably. That's when it went beyond strongly worded letters of disapproval.
If you want to talk about NATO provocations that weren't themselves in response to something Russia was doing, the only notable ones were missile defense proposals (eventually scrapped in part or whole with Obama's first year in office), and the enlargement of NATO in general. I call the latter a provocation only in the loosest sense, Russia was provoked, but it wasn't really a "wrong" thing to do. Post-soviet states were petitioning for NATO membership almost immediately, because of their desire to integrate tightly with Europe and guarantee their independence from Russia.
What they're designed for is less important than what Russia's strategic command needs to worry about. Another thing to note is that a lot of Russia's rhetoric isn't primarily intended for the outside world - it can well be an excuse for Putin to grab more power for himself.
Oh, there's definitely a lot of propaganda designed mainly for domestic consumption. Framing Russia as a state under siege by "the west" helps keep him popular even in times of economic trouble. It's an interesting balance of positions - Russia is simultaneously strong for taking action in Crimea and Syria, righteous for condemning military adventurism the west, but very concerned about the expansion of NATO.