Decklink duo, new version can send (monitor) and receive (record) signal, in setup card your select mode of work.
It's supposed to, yes.
By the time Support and I were done trying stuff and I had a few days to think about it, I'm pretty sure I did have the settings right, the UI is terrible, and I probably got a dud card.
Camera typicaly send only yuv422 via hdmi. Canon R300 is old camera, maybe you can change resolution to lower on hdmi output, sometimes you must change recording mode.
Another interesting point is that the monitor displays correctly when it's the only thing hooked up to the camera, but heavily magenta when it's running through the new AverMedia capture. But OBS using that capture with yuv422, is right. So I think the camera probably auto-selects which color space to use, and the monitor simply assumes RGB and displays accordingly.
It's not really a problem, because the camera has its own monitor built-in, and I normally drive the HDMI monitor from the computer anyway. Only for testing, or to satisfy an ID check, does it see the camera.
One of the reasons I got the AverMedia, in addition to the USB speed that you mentioned and the corresponding picture quality, was to deinterlace correctly and actually get a real 1920x1080 picture from the camera, which is its maximum resolution. It may be old, but it still works really well for what I do with it...except for the interlacing. Recording mode is as high-quality as I can get, because my SD cards are still big enough, and I use the recording for post-production. HDMI, of course, is for live and live-to-tape, both in OBS.