Print from the SD card and remove the auto reset jumper, then it doesn't matter if the USB disconnects mid print. Printing via USB isn't reliable on any reprap machine, that is why I include an SD card and a USB adapter in the kit.
If it disconnects as soon as you turn the bed or motors on that is always due to a poor ground connection to the Melzi. That connection carries about 15A so has to be very low resistance otherwise current will flow down the USB cable and corrupt the comms. If the bundle of wires is too big to fit the connector when tinned squeeze it flat with pliers or a vice to get one dimension small enough and then shave off solder with a sharp knife to get the other dimension correct. Trim the end flat. You then end up with a rectangular lump that fits the terminal block aperture.
If it resets due to mains interference you need to minimise the area of the ground loop caused by the two mains leads and the USB cable. That means plugging both the machine and the PC into the same wall outlet or the same extension lead and keeping the PC close to the machine with the short USB cable provided.
I can't see the pictures but Y drift is probably due to something unrelated to comms. Loose pulley or skipping steps. If it is the latter reduce the XY acceleration in the firmware from 4000 to 2000.
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