In the "Custom" tab of the Tascam Settings Panel (Not the Windows Sound Control Panel), you may need to raise the volume sliders and check the "Mic Monitor" & "Loopback" boxes.
If you want to hear both the input signal and the computer USB audio, you need to make sure the "Monitor Balance" knob on the hardware is turned to the halfway point. You may need to turn up the "Line Out 1-2" knob as well. In the Tascam Settings Panel, select the "Settings" tab. Make sure all of the boxes are checked to the right of "Audio Input ON", and select "Monitor Mix" for both "LINE OUT 1-2" & "LINE OUT 3-4".
In the "Settings" tab, if you only see options for "IN 1-2" in the "Direct Monitor" section, or lack any other channel 3 & 4 options in the "IO Settings", then you may have the drivers installed for the Tascam US-2x2 instead of the US-4x4. The Tascam Settings Panel V4.00 installer includes drivers for both the US-2x2 and US-4x4, so maybe there was an error during installation. If so, you'd need to uninstall The Tascam software and probably manually delete the leftover driver files from the "C:/Windows/System32/drivers" directory (For example, I have the US-16x08 and there's a leftover file titled "us-16x08.sys". In previous versions, there was also a file titled "Msft_Kernel_us-16x08_01011.Wdf"). WARNING, be careful not to delete anything other than the actual Tascam driver leftovers, because you could cause serious problems to your system otherwise. Just in case, check in Windows Device Manager for any more remnants as well.
Reboot, re-install the Tascam software/drivers, reboot again.
If it doesn't work after all that, I don't know what else it could be.