Experiments in low strain rate methane-air counterflow diffusion flames diluted with $CO_2$ have been conducted to investigate the flame extinction behavior and edge flame oscillation in which flame length is less than the burner diameter and thus lateral conductive heat loss in addition to radiative loss could be remarkable at low global strain rates. The critical mole fraction at flame extinction is examined in terms of velocity ratio and global strain rate. It is seen that flame length is closely relevant to lateral heat loss, and this sheets flame extinction and edge flame oscillation considerably. Lateral heat loss causes flame oscillation even at fuel Lewis number less than unity. Edge flame oscillations are categorized into three: a growing-, a harmonic- and a decaying-oscillation mode. Onset conditions of the edge flame oscillation and the relevant modes are examined with global strain rate and $CO_2$ mole fraction in fuel stream. A flame stability map based on the flame oscillation modes is also provided at low strain rate flames.