I'm working on a post-apocalyptic world for a game and the more I think about it the more unsure I get about the technological development of the setting being believable. 

So I'm asking how likely a certain level of technological regression would be given the following assumptions:

In a world that experiences a long lasting dramatic global cooling (a series of impact winters caused by a shower of asteroids that impact the earth over time for example, or maybe a volcanic winter?) for more than a century, reducing humanity to a population of maybe 2 million globally. How much technological regression would be plausible in this case? Assume that most technology that is in use by then has become complex enough that simply copying them cannot be done (requiring advanced electronic manufacturing and such).

* Would it be plausible if the knowledge of electricity is reduced so that it becomes impractical?
* Would it be plausible if the knowledge of chemistry was lost to such a degree that only the most useful and easiest to do formulas would be remembered?
* Would it be plausible if knowledge of manufacturing strategies were reduced to such a degree that humans would be stuck with artisanal production for a long time after?

Would it also be plausible if recovery was slow, in the level of several centuries, after such an apocalyse?

In such an apocalypse, global trade networks would collapse quickly I would assume (especially if there are armed conflicts between groups of survivors for the little fertile land that is still there during the long winter) I would assume and a lot of technology would be impractical for a long time in the future as requried materials would be very hard to get and might even be hard to salvage from the ruins of the old world in sufficient quantities. Though some of it would be possible I'm sure. Power sources would be limited, wind power might still be doable in the small communities that would survive but would it have enough practical uses to preserve for such a community? I would assume that most fuels would be impractical for such a community, with easily accessible oil and coal deposits already being depleted. So I guess easy portable electric power would be out of the question. Likewise, I also assume that the vast majority of people will be forced to do tasks that are required for basic survival, whether they work in Agriculture or other basic resource gathering operations.

Furthermore I would guess precision machining would not be doable without access to already complex machinery, limiting how complex any mechanical machines would be. But I'm unsure how fast these capabilities would be to reacquire. Finally, even if the old society had mostly phased out text books in favor of digital media would the surviving text books still around be enough to rapidly have society regain the lost knowledge to a level where they can make practical use of it?