[Aragorn]

As long as you are power cycling the modem between connecting/disconnecting the laptop, you should be OK. No need to release anything with IP config.
Can you confirm the cable between modem & router is plugged in to the WAN port on the router? How is the laptop connected to router - via a cable into one of the LAN ports? and you are able to access the router's web page? What IP do you get from the router, eg ipconfig while cabled into router, usually begins 192.168.x.x