Have you backed up all important data from the drive as a precaution.
Due to the folder behaviour I am not sure if it could be this but some LACIE devices have three physical power switch positions which are Off - Auto - On, if the device is normally used with the switch in the On position (permanently active mode) but has been accidently switched to Auto (power saving mode) position it could explain the device disconnecting itself.
I checked the power switch and it's only a On/Off switch.
Checking the USB cables and ports** + trying the LACIE drive connected to another computer to see how it behaves would be a good starting point.
I've checked the cables and there doesn't seem to be any problem, but I'm going to hook it up to my laptop and see how it does for the next couple of hours.
** The computer that the drive is connected to could be losing/turning off the power to the USB ports, again as a power saving measure or as a result of a possible fault.
I'll post back and let you know how it did with my laptop and we can go from there.