Star quad cable: Why does the white wire have noticeably less resistance

You didn't provide this information, so I am assuming your DAQ has differential inputs. If you have a sensor with a differential output, you may not want to load the two lines of the differential output differently - so they should go in a shielded twisted pair or two coaxes to maintain balance. Also, often you may not be able to connect one of those signals to your DAQ shield potential (most likely 0V) because of limitations on the sensor/driver side. That should answer that part.

If your sensor has a single ended output, you may very well connect a simple coax to your sensor and sensor-ground in one end and to your DAQ in_a and in_b/shield in the other end.

Just make sure you are not running serious current through that coax shield (due to different ground voltage potentials). If you have that situation, you need to ac-couple the connection - and thus would not be able to go all the way to DC. How much current in the shield is too much? I would set the limit at a value where your additional IR drop is around half a DAQ bit.

When you say signal is tied to the shield, which pins? The ground pins are tied to the shield. See pinout below

I am surprised to see that the grounds of the vga cable are tied to the shield, on the cables I have some of the grounds are tied to the shield. This is a more noisy solution, but with the cable making industry its about what works not what is best. They may be cutting corners by eliminating some conductors in the cable and running the ground through the shield.

A shielding conductor is used to prevent electrical fields to capacitively couple to the conductor they are protecting.

As far as turning the cable into an antenna:

From the cable shielding wiki:

The best method to wire shielded cables for screening is to ground the shield at ONE end of the cable

This is to avoid ground loops, ground loops are a big problem because current flowing through the cable causes current through the shield conductor to couple to the signals inside the cable magnetically. So to prevent this current flowing on the outside of the cable a choke can be placed on the outside of the shield to increase the high frequency impedance of the cable and cause high frequency signals to find a different "pathway" (as high frequency currents will "prefer" the path of lowest impedance/inductance).

The cable is shield is probably not tied to ground on both ends (computer and monitor sides), and one would hope that it is isolated somehow. Otherwise DC and AC currents will flow through the cable and cause problems.

So the shield shouldn't be tied to ground, it is probably introducing noise but it is and it still works, as far as we can tell. It would be interesting to find the VGA specs and see if this is a standard practice.