Why is the impedance zero at cutoff frequencies in a series RLC resonance circuit?
So does that mean that the impedance (Z = sqrt((Xl-Xc)^2 + R^2)) is
  zero at cutoff frequencies?


No, your math is wrong.

If \$|X_L - X_C| = R\$ at either of the 3 dB points (cut-off frequencies) then : -

$$Z=\sqrt{R^2 + R^2}$$

or

$$Z=\sqrt2 R$$