It really depends on what you mean by “play in 1080p”. No external video processor will be able to change the internal rendering resolution to 1080p, so they only way you’ll be able to play those games in 1080p in that sense would be through emulation.
If you mean play these games on a 1080p display, absolutely; you just need to feed your display a resolution it understands and let it scale up to its panel’s native resolution. With the OSSC, this would normally mean at least line-doubling 240p to 480p (most sets don’t accept 240p over HDMI), either passing-through 480i or bob-deinterlacing it to 480p, or passing-through 480p and letting the display handle the rest. If you want to line-multiply these to the maximum supported by the OSSC, you will need a display that can handle 960p over HDMI for 480i/480p content (line4x 240p, line4x 480i, line2x 480p) and/or deal with the top and bottom cropped on 240p material (line5x 240p, 1080p mode).
None of the devices that the OSSC would be used with support HDR. If what you’re talking about is that fake HDR that can be generated from SDR content, then I don’t see why not.
What do you mean when you say your games “speed up” when you’re not in game mode? Game mode is on the TV; it has no control over the framerate or processing on the console.