Predictions for the ability of 21 cm interferometric experiments to discriminate Epoch of Reionization (EoR) signal models are typically limited by the simplicity of data models, whereby foreground signals and characteristics of the instrument are often simplified or neglected. To move toward more realistic scenarios, we explore the effects of applying more realistic foreground and instrument models to the 21 cm signal, as well as the ability to estimate astrophysical parameters with these additional complexities. We use a highly optimized version of 21cmFAST, integrated into 21cmMC, to generate lightcones of the brightness temperature fluctuation for Bayesian parameter estimation. We include a statistical point-source foreground model and an instrument model based on the Murchison Widefield Array scaled in observation time to have an effective sensitivity similar to the future Square Kilometre Array. We also extend the current likelihood prescription to account for the presence of beam convolution and foregrounds, the two-dimensional power spectrum (PS), and the correlation of PS modes. We use frequency bands between 150 and 180 MHz to constrain the ionizing efficiency (ζ), the minimum virial temperature of halos (), the soft X-ray emissivity per unit star formation rate (SFR) (L X/SFR), and the X-ray energy threshold (E 0). We find that the inclusion of realistic foregrounds and instrumental components biases the parameter constraints, due to unaccounted for cross-power between the EoR signal, foregrounds, and thermal noise. This causes estimates of ζ to be biased by up to 5σ, but the estimates of T vir, L X/SFR and E 0 remain unaffected and are all within 1σ.