If you have a NIC that supports promiscuous mode is this possible? To my knowledge, I read that a switch uses a broadcast domain to broadcast packets, but if a router does not use this, is it still possible to sniff packets on your LAN?
Packets designated for the router will not propagate through the LAN network, at least not from the WAN side. All packets from inside the network will be available to other device as well, when put into promiscuous mode. A switch, which acts on MAC addresses, not on broadcast domains (thats a hub) can cause trouble indeed.
It is possible to sniff packets over LAN using a LAN Tap. These are passive devices that intercept traffic through the use of tools such as Wireshark.
However, these devices only capture traffic between two points, not always the entire network (as you may be used to with wireless packet sniffing). The devices uses three ports> Two of which, are passive and connect to the devices to the network, while the third port is for you to "tap" into the traffic between the two points.