Are all apps on Google Play compatible with the Samsung Galaxy S4? I wanted to know whether the Samsung Galaxy S4 will run every app from the play store or not? I had done a lot of googling but failed to find an answer.
In general, NO. 

Some apps:


run on specific devices only (i.e. not compatible with S4)
run on specific Android versions
available only in specific countries
available only on specific carriers
requires root access. (i.e. if phone is not rooted, it won't work)