This paper reports circular spectropolarimetry and X-ray observations of several polluted white dwarfs including WD 1145+017, with the aim to constrain the behavior of disk material and instantaneous accretion rates in these evolved planetary systems. Two stars with previously observed Zeeman splitting, WD 0322-019 and WD 2105-820, are detected above 5 sigma and <Bz> > 1 kG, while WD 1145+017, WD 1929+011, and WD 2326+049 yield (null) detections below this minimum level of confidence. For these latter three stars, high-resolution spectra and atmospheric modeling are used to obtain limits on magnetic field strengths via the absence of Zeeman splitting, finding B* < 20 kG based on data with resolving power R near 40 000. An analytical framework is presented for bulk Earth composition material falling onto the magnetic polar regions of white dwarfs, where X-rays and cyclotron radiation may contribute to accretion luminosity. This analysis is applied to X-ray data for WD 1145+017, WD 1729+371, and WD 2326+049, and the upper bound count rates are modeled with spectra for a range of plasma kT = 1 - 10 keV in both the magnetic and non-magnetic accretion regimes. The results for all three stars are consistent with a typical dusty white dwarf in a steady-state at 1e8 - 1e9 g/s. In particular, the non-magnetic limits for WD 1145+017 are found to be well below previous estimates of up to 1e12 g/s, and likely below 1e10 g/s, thus suggesting the star-disk system may be average in its evolutionary state, and only special in viewing geometry.
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