Many properties of the Milky Way's (MW) dark matter halo, including its mass-assembly history, concentration, and subhalo population, remain poorly constrained. We explore the connection between these properties of the MW and its satellite galaxy population, especially the implication of the presence of the Magellanic Clouds for the properties of the MW halo. Using a suite of high-resolution N-body simulations of MW-mass halos with a fixed final , we find that the presence of Magellanic Cloud-like satellites strongly correlates with the assembly history, concentration, and subhalo population of the host halo, such that MW-mass systems with Magellanic Clouds have lower concentration, more rapid recent accretion, and more massive subhalos than typical halos of the same mass. Using a flexible semi-analytic galaxy formation model that is tuned to reproduce the stellar mass function of the classical dwarf galaxies of the MW with Markov-Chain Monte-Carlo, we show that adopting host halos with different mass-assembly histories and concentrations can lead to different best-fit models for galaxy-formation physics, especially for the strength of feedback. These biases arise because the presence of the Magellanic Clouds boosts the overall population of high-mass subhalos, thus requiring a different stellar-mass-to-halo-mass ratio to match the data. These biases also lead to significant differences in the mass-metallicity relation, the kinematics of low-mass satellites, the number counts of small satellites associated with the Magellanic Clouds, and the stellar mass of MW itself. Observations of these galaxy properties can thus provide useful constraints on the properties of the MW halo.