Is the boot time of an iOS device affected by the apps that are running at shutdown? I suspect that apps that use system services might take time to register each app in the appropriate places at startup. 

Is the boot time of an iOS device affected by the apps that are running at the previous shutdown or the number of apps installed?
My experience is no one, ten or even 50 apps can slow down your boot in any measurable manner. Even an old iPhone 3 with 11 pages of apps booted in visually the same time as a freshly restored iPhone 3.

The reason for this is a number of design choices that provide real benefits to the running system and also are good at isolating boot times from application influences. Specifically, nothing auto-launches, there are not multiple user accounts to prepare, and the OS systems that display and interact with apps work from databases that update when apps are deleted or installed - not at boot time.