Can I run Wireshark on my Desktop PC?

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Can I run Wireshark on my Desktop PC?

Probably. It's probably running either Windows or Linux, and Wireshark runs atop both (although support for older versions of Windows has been dropped in newer versions of Wireshark).

My desktop PC is connected to my router via an Ethernet cable.

That should work, although the only traffic you're likely to see is traffic to and from the router (and traffic the router forwards on to the network to which it's connected and that it receives from that network) - you probably won't be able to capture traffic that's not directly going to or from your machine, or broadcast.