This paper investigates the combined benefits of full-duplex (FD) and cell-free massive multiple-input multiple-output (CF-mMIMO), where a large number of distributed access points (APs) having FD capability simultaneously serve numerous uplink and downlink user equipments (UEs) on the same time-frequency resources. To enable the incorporation of FD technology in CF-mMIMO systems, we propose a novel heap-based pilot assignment algorithm, which not only can mitigate the effects of pilot contamination but also reduce the involved computational complexity. Then, we formulate a robust design problem for spectral efficiency (SE) maximization in which the power control and AP-UE association are jointly optimized, resulting in a difficult mixed-integer nonconvex programming. To solve this problem, we derive a more tractable problem before developing a very simple iterative algorithm based on inner approximation method with polynomial computational complexity. Numerical results show that our proposed methods with realistic parameters significantly outperform the existing approaches in terms of the quality of channel estimate and SE.