HDMI is a video/audio standard, so I doubt it will help with USB.
When you say it takes forever, it really depends on the computer and your drive. If you don't have USB 3, then it will be slow for a 6 GB file.
When you delete something on the drive, do you also empty the recycle bin/trash? Because unless you set it to instantly delete it will pretend to delete the file but the space is still in use.
You could format the drive to ExFAT so that both sides can read/write but it still sounds like you are not fully deleting the file off the drive.
Two Macs can network at 10 Gbps over a direct thunderbolt link and the connection is relatively simple. Otherwise you (usually) need a crossover Ethernet cable and can network at 1 Gbps if the ports support it. This is the easiest way with just a cord, but it might be hard to set up a Windows to OSX connection.