  (Abridged) In 1999, Chandra revealed a 150"-radius X-ray halo surrounding the
40"-radius PWN G21.5-0.9. A 2005 imaging study showed that the halo is
limb-brightened, and suggested this feature is a candidate for the long-sought
SNR shell. We present a spectral analysis of G21.5-0.9, using the longest
effective observation to date (578.6 ks with ACIS, 278.4 ks with HRC) to study
unresolved questions about the spectral nature of remnant features, such as the
limb-brightening of the X-ray halo and the bright knot in the northern part of
the halo. The Chandra analysis favours the non-thermal interpretation of the
limb. Its spectrum is well fit with a power-law model with a photon index
$\Gamma$ = 2.13 (1.94-2.33) and a luminosity of L_x (0.5-8 keV) = (2.3 +/- 0.6)
x 10^33 erg/s (at an assumed distance of 5.0 kpc). An srcut model was also used
to fit the spectrum between radio and X-ray energies. We find that the maximum
energy to which electrons are accelerated at the shock ranges from ~60-130 TeV
(B/10$\mu$G)^(-1/2), where B is the magnetic field in units of $\mu$G. For the
northern knot, we constrain previous models and find that a two-component
power-law (or srcut) + pshock model provides an adequate fit, with the pshock
model requiring a very low ionization timescale and solar abundances for Mg and
Si. Our spectroscopic study of J1833-1034, the highly energetic pulsar powering
G21.5-0.9, shows that its spectrum is dominated by hard non-thermal X-ray
emission with some evidence of a thermal component that represents ~9% of the
observed non-thermal emission and that suggests non-standard rapid cooling of
the neutron star. Finally, the ACIS and HRC-I images provide the first evidence
for variability in the PWN, a property observed in other PWNe such as the Crab
and Vela.
