Preparing to PC couch experience
Using a shield for what you want to do is pointless and will also introduce latency, get a long HDMI cable and use the controller you have, if the PS3 controller experiences issues with lag/drivers/incompabibility just grab a Wireless 360 Controller for Windows from a PC store/Amazon and you will see that the 360 pad is natively supported in many games.
Just for the record, the speed of your internet connection has no relation with your ability to stream stuff on your own network, it only relies on the quality of your network, most likely in a home would be just your wireless router.
The information does not travel through the internet, it would go from the PC>Router>Shield>TV there is no outside stream, if your wireless router is up to the job of passing the information on fast enough and there is not interference causing issues you are good to go even if you have no internet connection at all.
Don't listen to anyone who says the speed of your internet connection matters, they don't know what they're talking about.

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