(Why) is it required to flash in order to install Google Play on CyanogenMod? Is it required to flash in order to install Google Play on CyanogenMod? If so, why? Or is there a method to install Google Play on CyanogenMod without flashing?

In comparison, other app stores such as F-Droid or aptoide can just be installed as APK, so I am wondering.

I am not wondering about why one needs to install CyanogenMod and the google apps separately, I know it is due to licensing issues.
As can be seen here, an answer has already been found. It appears that the only thing that justifies an installation as system app, for the Google Apps, is to make them unable to be uninstalled via the package manager (since you need root privileges to alter the System partition). As Firelord pointed out in the above answer, there appears to be no further difference in terms of functionality.

If you're wondering about why you're asked to flash CyanogenMod and the GApps separately, instead, I can respond that, due to license issues, the developers aren't allowed to include the GApps directly in a CM package.