OK. So the laptop is the ONLY PC in the house. And, you don't want to have to connect the external drive to the laptop? How is it going to function then? If you have a router and the external drive is capable of acting as NAS, then it could be done. Otherwise, it will need to be connected to the laptop or it has no access to the network.
IMO, firewall is not the issue. If it was, then you wouldn't connect whether wired or not.
I am assuming you're connecting the external drive to the laptop via a USB port?
Bryan