One of the major goals of the exoplanet community in the coming decades is to detect Earth-like exoplanets (exoEarths) and look for biomarkers in their atmospheres. High-dispersion coronagraphy (HDC) may allow detection and characterization to be done simultaneously, as well as relax the starlight suppression requirements of the telescope and coronagraph. However, similar to other direct imaging techniques, HDC faces challenging thermal and/or exozodiacal background levels. In this paper, we present simulations of coronagraphic observations using a variety of space telescope apertures ranging in diameter from 1 to 15 m, specifically incorporating thermal and exozodiacal background. We investigate the effects of instrument temperature and aperture on the maximum usable wavelength, as well as the effects of exozodiacal disk inclination and thickness on observational SNR. We then identify the spectral resolutions which maximize observational SNR subject to detector noise and the required starlight suppression levels for the detection of various potential biomarker molecules (H2O, O2, CO2, and CH4).