Is it possible to play games from my HTC Desire HD on my HD-TV?
Taking a look at the device specifications reveals:


The Sony Bravia KDL-55HX820 gives you four HDMI ports at your disposal according to TelevisionInfo.Com
The DHD doesn't have video out capability according to HTCDesireHDForum


So while your TV would support it, your DHD won't. Updating your device to any other Android version won't change that: the hardware does not support it (the DHD has no HDMI port, and its USB port is not MHL capable according to the information I found).

With that pre-condition broken, all the other questions are rendered meaningless. But for curiousity: Even if your Android device's screen can only display 400x800 resolution, the HDMI signal should carry the source resolution (and not the screen resolution). So if you e.g. try displaying 720p, it would be "rendered down" to 400x800 for your device's display, but if the HDMI connected device supports it fully, it would not be rendered down on the other end, to my understanding.