The initiation, propagation, and transition of the autoignition assisted spherical cool flame and double flame are studied numerically and experimentally using n-heptane/air/He mixtures under shock-tube experimental conditions over a wide range of temperatures. The primary goal of the current study is to understand the effects of the ignition Damkohler number, ignition energy, flame curvature, and autoignition-induced flow compression on the propagation of spherical flames to ensure the proper interpretation of shock-tube flame speed measurements at engine-relevant conditions. The results show that at high ignition Damkohler number, there are three different flame regimes, cool flame, double flame, and hot flame. The cool flame speed accelerates dramatically with the increase of ignition Damkohler number. In addition, it is found that the change of flame regime, low-temperature autoignition, flame stretch, and autoignition-induced flow compression result in a complicated non-linear dependence of flame speed on stretch. The results also reveal that the spherical cool flame has much lower Markstein length compared to the hot flame at T > 600 K. Moreover, it is found that both the autoignition assisted cool flame and the trailing hot flame front in the double flame can propagate much faster that the hot flame alone at the same mixture conditions, leading to a nonlinear dependence of flame speed on the mixture initial temperature. The simulated flame trajectories and the flame speed dependence on temperature agree qualitatively well with the shock-tube experiments. A quantitative criterion to ensure the accurate speed measurement of the cool and hot flame is proposed. The present study provides important physical insight and guidance for the flame speed measurement using a shock-tube at engine relevant conditions.