This thesis describes the results of two separate analyses. Part I is the description of the first analysis which uses the newest measurements of neutrino mixing to study various non-standard models of neutrino interactions through their impact on solar neutrinos. These models can be motivated by the fact that solar neutrino experiments have yet to see directly the transition region between matter-enhanced and vacuum oscillations. The transition region is particularly sensitive to models of non-standard neutrino interactions and propagation. I examine several such non-standard models which predict a lower-energy transition region. I find that while several models provide a better fit to the solar neutrino data set, large experimental uncertainties lead to a low statistical significance. Part II describes the second analysis, where I look at neutron followers of contained atmospheric neutrino events in the SNO data set. These kinds of events are difficult backgrounds for nucleon decay measurements, and understanding the neutron follower multiplicity will allow for better rejection. It can also help improve measurements of the neutrino mass hierarchy and neutrino-nuclear cross sections. I find that the dependence of the average multiplicity on the visible energy agrees well with the predictions of simulations except for an unexplained deficit between 100 MeV and 600 MeV and an excess above 4 GeV. I determined the ability to distinguish neutrino and antineutrino events using the multiplicity by fitting for the double ratio $R \equiv (\overline{\nu}/\nu){\text{data}} / (\overline{\nu}/{\nu}){\text{MC}})$. I find $R = 0.93^{+0.91}{-0.63}$ for a fit to a single multiplicity distribution per phase, and $R < 1.00$ for a fit to separate distributions for single electron ring, single muon ring, and multi-ring events. I also look at the agreement with a meson-exchange current cross section model developed to explain anomalous cross sections measured by MiniBooNE. Fitting for the strength of the MEC contribution as a fraction of the quasielastic charged-current cross section, I find an upper limit of $\sigma{MEC}/\sigma_{QECC} < 0.17$ for a fit to combined distributions and $\sigma_{MEC}/\sigma_{QECC} < 0.04$ for a fit to separate distributions for ring count and type.