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Is the hardware I plan to purchase sufficient to solve the task?

Not enough information.

Does OpenCV support reading video feed from this capture card/HDMI input?

Almost certainly not, but if you can get a pointer, you can construct an OpenCV Mat around it and everything is fine.

Should I focus on using Linux or Windows, what platform is better supported by OpenCV? (hopefully not MAC)

Either or. Both are well-supported. Your comfort with it makes much more difference.

Where/How should I start with the programming?

Start with the Python Tutorials. Get comfortable with OpenCV, learn how it works ect. Then the Image Stitching tutorial. It's in C++, so you'll need the previous tutorials to understand how to translate it to Python. The rest of what you need is in the documentation on Stitching. The basic tutorial is not for video, but by the time you've worked through them all, you'll know how to read the documentation and have a better idea of what to do to make it fast for video.

Good Luck.