Can I use target disk mode using an USB 2.0 to Firewire cable?
I can't find a specific Apple KB reference, but the consensus seems to be that you need Firewire, Thunderbolt or USB-C; USB 2 doesn't appear to be supported.

From the MacSales blog - Mac OS 101: How to Boot and Use a Mac in Target Disk Mode


  What You Need
  The list is short, but essential.
  
  
  Two Macs. That may seem obvious, but it makes sense to point out that Target Disk Mode only works between two Macs; you can’t chain multiple Macs together. All of the connection types (FireWire, Thunderbolt, and USB-C) support hot connecting, meaning you can connect a cable between the two Macs while they are powered on. We recommend shutting down both Macs before proceeding, however.
  An appropriate cable to make the connection. Ideally, you should connect similar ports; that is FireWire to FireWire, Thunderbolt to Thunderbolt, or USB-C to USB-C. There are, however, exceptions. Using adapters to connect Thunderbolt to FireWire will usually work, as will Thunderbolt to USB-C. But not all adapters are known to work correctly in Target Disk Mode, so if you can, connect directly to the same port type.