How modern messengers encryption keys can't be hacked I dont know is this question more like reverse-engineering or cryptographic.
Why governments are hunting for encryptions keys? Aren't they delievering by the network the same as the encypted messages? What doing it end-to-end encrypted?
Aren't they delievering by the network the same as the encypted messages?


No. At least not in the plain. 


  What doing it end-to-end encrypted?


Try to search for the key agreement protocol (e. g. Diffie-Hellman ) and asymmetric encryption (e.g. RSA ). 

The "asymmetric" encryption allows sending encrypted and signed messages without sharing the private (secret) key. 

Very simplistically explained - each actor creates a key pair (with a public and private part) and the actors will exchange their public key. That allows both actors to securely create a shared secret even on an unsecure channel