For the USB to HDMI route, you have a few options. Samsung recently switched to a 11 pin USB connector (I believe the Galaxy S3 was their first phone with the change). The Note 2 has the same 11 pin connector. You can either buy the new MHL main adapter with 11 pins.
HDTV Universal Adapter (11 Pin) ) or you can use an older MHL adapter like the one you linked to and buy a 5 pin to 11 pin adapter like this Samsung one
HDTV Adapter TIP (5 to 11 Pin Converter) I had a MHL main adapter (5 pin) that I bought originally for my Galaxy S2. I bought the 5 pin to 11 pin adapter from Monoprice and can confirm all is well.
For only $3.57 each when QTY 50+ purchased - 5pin to 11pin Micro USB Adapter for MHL Adapters and Samsung Galaxy S3 | MHL Adapters
Your other option (the cheapest if your TV or game box is DLNA compliant and connected to wifi/ethernet) is to stream the video over wifi. I have tried this with my PS3 and GN2. I tried turning on the Allshare feature, but couldn't get my PS3 to see my phone. Then I tried a share media option that was near the Allshare option in the settings and was able to get the Note2 to show up on my PS3. I then played part of a movie. I don't know what resolution is supported by DNLA/streaming, but from what I saw it didn't look as high definition as the original file (I was streaming a bluray rip that I had stored on my Note2).