how to lower reactance of an electromagnet?
If you are driving the coil with (say) a transistor you will have a diode across the coil to prevent back-emfs destroying the transistor however, this makes the turn off time quite long so you might consider, as a simple option, using a zener diode in series with the protection diode. This burns the stored energy much more quickly.


  If I were to bundle 4 strands of magnet wire and coil the bundle 500
  turns (500 x4 =2000 turns) and then connect each wire in parallel,
  then adjust the voltage so that the current in each wire is still 0.6,
  would that improve the response time of the magnet (by lowering the
  impedance and therefore reactance of each individual coil) ??


You have to keep ampere turns the same for producing the same mechanical force so, 500 turns x 2.4 amps = 2000 turns at 0.6 amps. In other words, it's a bit simpler than what you propose but you can do it your way to re-use the wire.

Force = \$(N\cdot I)^2\cdot 4\pi 10^{-7}\cdot \dfrac{A}{2g^2}\$


F = Force
I = Current
N = Number of turns
g = Length of the gap between the solenoid and the magnetizable metal
A = Area of solenoid/electromagnet


Reducing turns by a factor of 4 (irrespective of number of parallel windings) reduces the inductance by 16 (ratio of turns squared) so your response speed improves by a factor of 16. This assumes the same core and geometry of windings on that core (which I'm sure in this example will be true). So if you only wished to quarter the inductance (1/4 the response time) then halving the turns to 1000 would work and only requires 1.2 amps.