Yes, you can play games at full screen mode for pretty much any resolution. (hell, most games lets you "stretch" 800x600 or 1024x768 resolutions to fit the 16:9 screen (making it look stretched out in the process lol)
The problem is that when you play at lower resolutions, it can led to pixel stretching (like one pixel taking up two or three physical pixelso n the screen)
EX: if you're playing at 720p on the 1080p resolution, more jaggies will appear since there are roughly 360 vertical and 640 horizontal lines being stretched to cover the entire screen.
So basically it would look worse than a 720p monitor of the same size if you play on 720p on 1080p monitor...
and by downscaling, you mean upscaling a lower resolution to a higher resolution screen.