Protecting Crimea from Ukraine can currently be accomplished by the Coast Guard... As for Japan and the Kurils, this is where a Russian CVBG in the north Pacific would suddenly matter a whole lot. It's one thing to plan an amphibious operation against the Kurils where your biggest threat is some aging Soviet destroyers, a handful of subs, and a few new frigates/corvettes. It's another when there's a Russian CVBG out at sea that could strike at any time. At the very least considerable resources would have to be devoted to tracking it and being ready to counter it. At worst it could make the entire operation unworkable. This is exactly what I meant when I was talking about a headache for anyone planning against the VMF.
In my opinion the main purpose of the Russian Navy, in order, should be as follows;
1) protecting SLBM bastions from foreign ASW
2) protecting Russia's ports and coastlines from enemy attack
3) protecting and providing communication/supply lines to Russian outposts (Kaliningrad and Kurils, but also Syria and soon Sudan)
4) protecting Russia's claim to the Arctic and the Northern Sea Passage
5) protecting Russian trade routes where feasible (this would range from something like anti-piracy missions, to protecting Russian tankers going to Venezuela in open defiance of US or EU sanctions)
6) providing support for Russian expeditionary forces