Are we talking about doing all of this on the phone? If so, the HTC One X5 is the only phone that is even rumored to have a resolution of 1080p.
If your're looking for raw power, the LG Optimus G is the way to go, as it will pack Qualcomm's new S4 Pro processor, which is roughly double the performance of the Tegra 3 and Tegra 3+ that will be in the HTC One X5.
The Note 2 is a multimedia device. It'll have a nice processor and a standard 720p resolution (which might look a bit more pixelated splayed out across 5.5"), but unless you're looking for a huuuuge screen, I wouldn't go with it.
Their aren't enough "hardcore" games on the Lumia 920.
The iPhone wouldn't be a bad bet, but if you're expecting to stream content from your phone to your TV, that won't be happening, and the resolution won't be better than ~any~ of these other phones. However, seeing as Apple integrates their software and hardware in a way that no company is able to do, the games are ~very~ optimized. The same game on an iPhone and an equivalently powered Android will be smoother on the iPhone.
The S3 is just a culmination of the standards of the smartphone industry. If you're going to consider it out of all these, I'd recommend the HTC One X instead, they both use the same processor in the US, and I love HTC's designs.
I really don't know anything about Sony's phones, apologies
Personally, out of all of these, I'd get the Optimus G, just because you said you wanted to play hardcore games. That will last you more than the two years of your contract before you can upgrade again