Yes, but not using that tool. Certainly not at a $8 price-point over USB 2.0 (and while some of them have USB 3.0 looking ports, they're USB 2.0 internally).

I'd start by looking at the Black Magic Intensity Pro 4K, but you're looking at hundreds of dollars.

With USB2.0 you're talking about 53MBytes/sec data rate. That's more than sufficient for lossless Full-HD video.

Yes, for anything higher quality (UHD 4K) you need more expensive hardware.

I have multiple of what I'm very confident are basically those devices. I use them fairly regularly.

They are almost all built around the MS 2109 Macrosilicon chip (and the ones that aren't are worse). They will only do MJPEG when above a certain rez and framerate.

1080p60 24-bit colour depth is 372.5MBytes/sec uncompressed. So it's going to require some compression.

I'm not an expert on h.264 etc, but can you do real-time lossless CBR below 53MBytes/sec on a readily available hardware encoder that, including the HDMI capture functionality, comes anywhere near $8?