Is the external hard drive connected directly to your Mac using USB or FireWire, etc.?
Is the external drive formatted in Mac OS Extended (Journaled)
Did you Open System Preferences (gear icon on the dock), open Time Machine, click Time Machine Options, click the drive to highlight it, then click the - (minus) button to remove the drive from the list of Excluded Items?
If yes, the drive is being backed up and you can see it when you open Time Machine under the DEVICES heading on the left side of the Finder window.
Open the drive to see that data this is being backed up.
I have not seen the "shaky" mouse behavior here, so I do not have a solid answer.
Wireless mouse? Are you backing up the external drive using wireless?
If yes, it sounds like there may be some interference effects at play here. Try restarting the wireless router when you can, and it might change channels, so the devices are not in conflict during large data transfers.
Or, manually change the 2.4 GHz channel on the wireless router when you can to see if that helps.