How to remove a Virus that installs an unlimited amount of APPs causing the smartphone to stop once an internet connection has been established?
As I already mentioned in the comments, here are the primary points you need to be careful about:


Whenever you install an app from the Play Store, you should pay attention to things like permissions, ratings, reviews by existing users, etc. before installing it on your device. One malicious install, and your phone could become compromised. It doesn't matter then, whether you are rooted or not.
When you install an app, it asks you a bunch of questions like what all things it needs access to (like SDCARD, telephony resources, system tools, etc.). Even without root access, a malicious app can do lots of damage to your phone, if you permit them these resources.
As the existing answer suggests, rooting is your only option if the malware has landed itself in the system area. But if you go along that route, proceed with care and caution, as there is a risk of bricking if incorrectly done.
For future reference, make a mental note to double check at least two things before installing an app: 1) The permissions it needs are sane and safe. 2) There are at least 500 million installs already with at least a 3.5+ rating. You can make rare exceptions for popular apps like WhatsApp/Skype, but this is generally what you should do. You may or may not be able to recover this phone, but life goes on and these things will go a long way to ensure that such a situation will never arise.