Can I install CyanoGenMod's or AOSP Messaging app on top of existing Messaging app by my vendor I want to install default Android SMS/Messaging app by Google, or even the one that comes with CyanoGenMod Custom ROMs.
They feel more native and plain, which is awesome.  

Can I have 2 Messaging apps on my Galaxy S4 (I9500) without problems, or disable Samsung's messaging app and install AOSP/CM's Messaging app?
Starting with Kitkat, you can make one app the default SMS app. By default, on a device with Google apps, that is Hangouts, but you can install as many SMS apps as you like. If they have been developed/updated for Kitkat, they should be fitted to be used as "default SMS app".

Of course, if you don't need one of the pre-installed apps, or don't want to use it at all, you can go to Settings→Apps and see whether it can be disabled (some pre-installed bloatware is protected against this).