I bought the router (the one, I mentioned above: D-Link DIR-615). I tried to reconnect the way I used to did before with my modem, but it didn't work.
So I created a reconnection script with the Reconnect Recorder and it's perfect. The only thing is: this way (with the use of a router+modem instead of a single modem) of reconnection is a little bit slower (it takes about 40 seconds the router to restart and change the IP), in front of the modem reconnect 15-20 seconds. However during the creation of the script I used as few clicks as possible: on my router's web interface there's an "Internet Online" sign with a "Reboot" button under, I clicked on it.
So thank you for the support, I can keep using and enjoying this perfect download manager.