Cyanogen and Google Play compatibility I already read, how to install Google Play on Cyanogen. But then what? Do the apps from Play store install seamlessly, like on "standard" Android? 

Some apps require minimum version of Android. Google Play "tells" me, if certain app is compatible with my device. If eg. app requires Android 4.4, does it mean, that it will or will not work on Cyanogen?
The guide is only to install Google Play and other related and necessary Google Apps. The installation is such due to licensing issues which prevent CyanogenMode from packaging it in the ROM directly.

Once you go through the install of Google Apps, you can use Google Play same way as on any other Android device.

Regarding versions, every CyanogenMod version is based on particular version of Android, e.g.:


CyanogenMod 11 is based on Android 4.4 KitKat
CyanogenMod 10.2 is based on Android 4.3 Jelly Bean
CyanogenMod 10.1 is based on Android 4.2 Jelly Bean.
etc.


The play store will treat CyanogenMod versions as Android of appropriate version, so you will have no problems regarding that.