The Epoch of Reionization (EoR) is the period when the universe transitioned from neutral to ionized, as the first stars and galaxies reionized the hydrogen atoms in the intergalactic medium. This period is challenging to observe directly and remains one of the most exciting frontiers in modern astrophysics and cosmology. The 21 cm hydrogen line arises from the spin-flip'' transition in the hyperfine structure of hydrogen atoms, which occurs due to differences in energy between the two states when the spins of the electron and proton are parallel or antiparallel. This line serves as a direct probe for tracing the distribution of neutral hydrogen in the universe and is a promising tool to provide us with a comprehensive three-dimensional evolutionary picture of cosmic reionization. With various ongoing and upcoming radio telescopes targeting the 21 cm signal from EoR, precise measurements of the 21 cm brightness temperature power spectrum are crucial. The 21 cm cosmological signal suffers from contamination by much brighter foregrounds. To address this, the Hydrogen Epoch of Reionization Array (HERA) utilizes the delay spectrum approach to measure the power spectrum, aiming for a foreground-avoidance'' strategy. By dividing the delay space into distinct foreground-dominated'' and noise-dominated'' regions, a unified error estimation pipeline becomes essential for robust measurements. As supporting material to published HERA upper limits, we conduct a critical examination of different error estimation methodologies available for 21 cm delay power spectrum measurements in HERA. This involves synthesizing analytic work, simulations of toy models, and tests on small amounts of real data, and comparing results across different methods. We find that different error bar methodologies, although computed independently, are in good agreement with each other, and also demonstrate the advantage of the methodology used for error bars in the HERA public data release. The cosmological 21 cm signal is thought to be unpolarized, and the 21 cm power spectrum is measured from the total sky intensity, specifically the Stokes \textit{I} field of the sky. Due to miscalibration and instrumental effects, polarized foregrounds can leak into the unpolarized measurement, a phenomenon known as ``polarization leakage''. If we cannot disentangle the polarized components, we risk misestimating the true total intensity, which could hinder our ability to cleanly measure the EoR signal. Addressing this issue requires fully polarized calibration, which is currently lacking in the HERA data processing pipeline. We test the Smirnov-Tasse algorithm to apply further polarized calibration to a small part of the HERA-19 commissioning array dataset, significantly improving the characterization of polarization leakage compared to the original simple image-based calibration. There is still ample future work needed to test this method on more recent HERA data. With improvements in sensitivity and analysis techniques, precise 21 cm measurements hold great potential for cosmology. The most important cosmological parameter that 21 cm observations can measure directly is $\tau$, the Thomson-scattering optical depth of CMB photons due to free electrons from reionization. This is one of the six base parameters in $\Lambda$CDM. However, in CMB measurements, it is degenerate with another important parameter, $A_s$, the amplitude of primordial density fluctuations, which weakens the constraints. The potential for EoR 21 cm measurements to accurately measure $\tau$ and treat it as a nuisance parameter when extracting other cosmological parameters from CMB is an exciting prospect. We focus on methodologies for inferring $\tau$ from 21 cm observations. Previous works have demonstrated a method that first forecasts astrophysical and cosmological parameters from the evolution of the 21 cm power spectrum, and then runs a simulator to produce $\tau$ once those underlying parameters have been determined. However, the path from power spectra to $\tau$ is not straightforward, as it relies on the specifics of intermediate simulators, posing a risk of model-dependent interpretation. We extend these works by testing the consistency among various simulators in recovering $\tau$ from the same mock data, which includes noise, redshift coverage, and wave-numbers appropriate for HERA measurements. We also examine which wave-numbers and redshifts contribute the most to the constraints on $\tau$, enhancing our understanding of the connection between the evolution of power spectra and $\tau$. This will help achieve more robust cosmological measurements from 21 cm data.