Why does Android check email and play store when sync is off At night, or at work, my Android is usually offline. For me it's enough to only check email when I like to do so.

However, every time (or most times?) I switch on mobile data on wifi on my phone, it also seems to sync email (corporate Exchange account only, NOT the gmail app), and also checks for Play store updates for my apps. Same goes for Cyanogenmod updates.

Why does it do so? What's the use in turning off Sync, when apps can just ignore the setting and run background checks whenever they like?
I think you've misunderstood the sync settings. They're not intended as a way to stop apps performing background operations. The sync settings for each app come from within that app, like any other app settings, so the exact behaviour is up to that app. Because the sync settings are part of the accounts mechanism, they're usually used to control data attached to that account (e.g. if you have the Evernote app installed and an Evernote account signed in, the sync setting within that account just controls syncing the data owned by that account; it doesn't control any other kind of background processes).

OS and app updates aren't covered by any of the sync settings, so you wouldn't expect those settings to change their behaviour.