^^^^BUY ONE OF THESE ^^^^
I bought one and have really enjoyed using it. This box plays everything I have thrown at it where the others (roku, Chromecast, smart TV, etc...) have failed. I plan to buy at least two more. One to replace a HTPC that I built and was planning on spending upwards of $600 to upgrade the processor. No need now with a <$90 android box.
The stock browser will pay the NCHC games fine. And can download any other browser if needed via the Google play store. It even has XBMC installed for a nice interface. Although kind of complicated to use and set up.
It comes with a remote. But I downloaded droidmote server on the box which is $2.99. And the droidmote client on our phones (free). Allows you to control by a mouse, keypad, and even a game controller. Can even use the phone popup keyboard (swipe) for faster text entry in browser usage.
I'm not sure about the big sky or whatever else games. But if it has flash or any other current html5 it will play in the stock browser. If not, then try Firefox or chrome. Something not even apple TV has!
I just wish there was a easy kiosk mode or login for guest on the NCHC so I can let my parents use without access to account stuff.