If you want to stream Dolphin to your TV, a cheap option is to use Steam's in-home streaming feature. The Steam Link is a dedicated piece of hardware for this which I think is about $50 and will work with existing controllers or the Steam controller. You can also just use any existing laptop (windows, mac, linux) with a sufficiently powerful cpu for stream decoding and use it as an in-home streaming client. I've messed around with a lot of solutions for streaming from my PC to living room TV, and Steam's solution blows just about everything else out of the water.
From my personal experience (and research online), wifi can be iffy, and ethernet ensures a much smoother experience overall.
Though maybe my experience is affected by the router I have. (which is in AP mode and I have a custom firewall)
Ethernet FTW.