I need the HDMI cable for other uses but I'm currently downloading something. If I unplug it will it stop the download?
The HDMI cable is merely an output to an external display which a console without an inbuilt display needs to display content.
In other words, YOU need it to see what the Xbox is doing, but the Xbox has no need of it. So if you remove the HDMI cable, the Xbox will simply stop displaying, which is not necessary to continue downloads.
As long as your internet connection is solid and you have uninterrupted power, you're golden.