Video card, I recommend the Radeon 6870 because it can handle 1080p gaming well so it should be perfect with your lower monitor resolution(s).
The i7-2600K will not be any faster than the i5-2500K in gaming today but maybe it will in a few years. Even if it will the 6870 isn't fast enough to be bottle necked by the i5-2500K. The i7 would be a more future proof solution in the long run but nothing more than the i5-2500K in the short term of one or two years.
Multimedia streaming doesn't even use up as much processor performance as the 6870 does in gaming so the i5-2500K would be overkill if not for the gaming aspect, let alone the i72600K.
Multimedia streaming on a network is not a demanding task but gaming is so the parts would be geared towards gaming. This will not hinder the streaming usage of the computer but it will increase the cost to get parts that can handle gaming.
A PCIe wireless adapter would be your best option for a media streaming machine because PCIe can offer better performance than any PCI and/or USB adapter if you get one of the right adapters. I don't know about tv tuners because I've never had one.
For keeping your system quiet you would need a video card with a quiet cooler and you would need an aftermarket CPU cooler. A good yet not expensive cooler would be the Cooler Master Hyper 212 Evo priced around $30. you also need a quite PSU because some of them are loud enough to hear vividly from a few dozen feet away.
I don't claim to be an expert on networking but I would think that a bad ping is more related to your internet connection than your computer unless you have a very poor quality network adapter.
For gaming I now recommend having 8GB of RAM in a 2x4GB kit running at 1600MHz 9-9-9-24 or better. 1333MHz memory is about the same price so the negligible performance increase by going to 1600MHz is more than justifiable and anything less than 8GB could limit some games. besides, even that kind of kit would only cost $35-45 or so so skimping on memory is pointless.
As for the case, I just use the absolute cheapest case that fits my needs but I often skimp on cases because spending more on a case won't increase performance unless you are using components that put out a lot of heat. My last case was an ATX mid-tower with 7 expansion slots, 4 5.25" bays, 4 3.5" bays, and a 2.5" bay and it only cost me $40 including shipping. It even came with a cheap 480w PSU. it was a used case but I got a very cheap case and the only problem was a poorly placed bolt that made using the second expansion slot slightly more difficult.
Point is with your setup you can consider a cheaper case than your choice.