So i can give my own post some answears
while i was writing the question the answear appears on my mind... i forgot to deactivate the echo mode of my modem (ATE[<value>]) so now i was able to receive only the response od my modem.
But still i havent fixed a problem when i first power up my modem... i have to write the first at command via screen /dev/ttyUSB0 115200 to the modem before i can run my c++ binary and read and write commands right.
Also i'm still not happy with my serial class, i dont know if my approch is the best way to do this. Mybe someone have some more experience and can give me some hints ?