We present the results of a deep Chandra observation of the X-ray bright, moderate cooling flow group NGC 5044 along with the observed correlations between the ionized, atomic, and molecular gas in this system. The Chandra observation shows that the central AGN has undergone two outbursts in the past 100 Myrs, based on the presence of two pairs of nearly bipolar X-ray cavities. The molecular gas and dust within the central 2kpc is aligned with the orientation of the inner pair of bipolar X-ray cavities, suggesting that the most recent AGN outburst had a dynamical impact on the molecular gas. NGC 5044 also hosts many X-ray filaments within the central 8kpc, but there are no obvious connections between the X-ray and H$alpha$ filaments and the more extended X-ray cavities that were inflated during the prior AGN outburst. Using the linewidth of the blended Fe-L line complex as a diagnostic for multiphase gas, we find that the majority of the multiphase, thermally unstable gas in NGC 5044 is confined within the X-ray filaments. While the cooling time and entropy of the gas within the X-ray filaments are very similar, not all filaments show evidence of gas cooling or an association with Ha emission. We suggest that the various observed properties of the X-ray filaments are suggestive of an evolutionary sequence where thermally unstable gas begins to cool, becomes multiphased, develops Ha emitting plasma, and finally produces cold gas.