Of course the salary cap has forced them into making a whole lot of changes this summer but before that, I wouldn't say there was anything special about their activities.
Players leaving to the NHL are inevitably going to happen and even of those you have listed, none did leave at the same time. I mean there is a 2 year gap between Dadonov and Gusev. It's just the way SKA has chosen to operate, much like how they hold rights for most of the Russian guys still in NA now. Basically they went "there is no way to avoid losing players to NHL but all the distribution of those locally are going to go through us". They will be able to decide if they keep the guys who want to come back to the K (like Soshnikov who has never played a game in SKA system and yet Salavat had to trade Tkachyov for him) or let them go elsewhere. And same applies to the current team, everyone understands Podkolzin, Marchenko, Morozov, Kuzmenko are only there for 2-3 more years, at best.
Regarding the past results, CSKA and SKA were always going to in their own 2 horse race and CSKA had the upper hand now with Kaprizov, Grigorenko, Sorokin. High-end Russian players are few and far between and SKA have missed with their picks. Datsyuk didn't really deliver for the role, Yakupov was slotted as Dadonov's replacement and was virtually useless. But if Kaprizov and Sorokin leave it will once again reset the playing field and SKA already lured Vey over to them. CSKA is currently going through a bigger shock than SKA ever did, especially if Grigorenko leaves as well. SKA, for the most part, was in the situation where they have to replace 1 top player per season, CSKA are in the situation where they will be without entire first line.
And yes, they didn't have a lot of success with coaches post-Znarok (although Kudashov's results last year were by no means bad) but at the same time, it might seem odd for an NHL fan, but "a new season - a new coach" practice is rather common in the KHL. SKA managed to avoid mid-season firings which is, in itself, a sign of relative stability.
I think they need to add a high-end right wing (maybe ditch Kemppainen to do it) but other than that, they are a perfectly well-built team. At this point, it's 2 tiers above what CSKA currently has.
Not exactly sure how the arena is impacting them in any way. It's not like money is, was, or would ever be a concern for them. The salary cap isn't adjusted to the seats in the arena either so
Pretty irrelevant to their fortunes on ice if you asked me. You know as well as I do every Russian team is reliant on someone's charity and they see no issue with it.