Given that this was originally asked on Information Security, rather than, say, Network Engineering, when you ask "Can I sniff the packets being sent to and from the wireless router even though I am connected through cable?", what you might really mean is "could somebody sniff the packets being sent to and from the wireless router...".
If so, then the answer is "yes"; if they're within radio range of the wireless router, and have a Wi-Fi adapter that they can put into monitor mode, they can sniff traffic on your network.
However, if the network is "protected", i.e. using WEP or WPA/WPA2, the traffic is encrypted, and they'll either need the network's password or software capable of cracking the packets to decrypt it. WEP is definitely crackable, so it's Equivalent Privacy to that of a Wired network only if the wired network to which it's being compared isn't hard to cut into and tap. :-) WPA and WPA2 are harder to crack.
If, however, you really were asking whether you could capture the traffic, the above applies. You're running an operating system that does let you put adapters into monitor mode. For various annoying reasons that I hope to have time to fix in libpcap at some point, it's not as easy as it should be (i.e., the checkbox in Wireshark and the
-I flag in tcpdump and tshark don't necessarily work), but it can be done.
As for decrypting the packets, here's how to decrypt them in Wireshark. Note that for WPA/WPA2, you need to capture the initial EAPOL handshake for each station whose traffic you want to decrypt.