In-band full-duplex (FD) operation is practically more suited for short-range communications such as WiFi and small-cell networks, due to its current practical limitations on the self-interference cancellation. In addition, cell-free massive multiple-input multiple-output (CF-mMIMO) is a new and scalable version of MIMO networks, which is designed to bring service antennas closer to end user equipments (UEs). To achieve higher spectral and energy efficiencies (SE-EE) of a wireless network, it is of practical interest to incorporate FD capability into CF-mMIMO systems to utilize their combined benefits. We formulate a novel and comprehensive optimization problem for the maximization of SE and EE in which power control, access point-UE (AP-UE) association and AP selection are jointly optimized under a realistic power consumption model, resulting in a difficult class of mixed-integer nonconvex programming. To tackle the binary nature of the formulated problem, we propose an efficient approach by exploiting a strong coupling between binary and continuous variables, leading to a more tractable problem. In this regard, two low-complexity transmission designs based on zero-forcing (ZF) are proposed. Combining tools from inner approximation framework and Dinkelbach method, we develop simple iterative algorithms with polynomial computational complexity in each iteration and strong theoretical performance guaranteed. Furthermore, towards a robust design for FD CF-mMIMO, a novel heap-based pilot assignment algorithm is proposed to mitigate effects of pilot contamination. Numerical results show that our proposed designs with realistic parameters significantly outperform the well-known approaches (i.e., small-cell and collocated mMIMO) in terms of the SE and EE. Notably, the proposed ZF designs require much less execution time than the simple maximum ratio transmission/combining.