This paper studies integrated sensing and communication (ISAC) with dynamic time division duplex (DTDD) cell-free (CF) massive multiple-input multiple-output~(mMIMO) systems. DTDD enables the CF mMIMO system to concurrently serve both uplink~(UL) and downlink~(DL) users with spatially separated \emph{half-duplex~(HD)} access points~(APs) using the same time-frequency resources. Further, to facilitate ISAC, the UL APs are utilized for both UL data and target echo reception, while the DL APs jointly transmit the precoded DL data streams and target signal. In this context, we present centralized and distributed generalized likelihood-ratio tests~(GLRTs) for target detection treating UL users' signals as sensing interference. We then quantify the optimality and complexity trade-off between distributed and centralized GLRTs and benchmark the respective estimators with the Bayesian Cramér-Rao lower bound for target radar-cross section~(RCS). Then, we present a unified framework for joint UL users' data detection and RCS estimation. Next, for communication, we derive the signal-to-noise-plus-interference~(SINR) optimal combiner accounting for the cross-link and radar interference for UL data processing. In DL, we use regularized zero-forcing for the users and propose two types of precoders for the target: one ``user-centric" that nullifies the interference caused by the target signal to the DL users and one ``target-centric" based on the dominant eigenvector of the composite channel between the target and the APs. Finally, numerical studies corroborate with our theoretical findings and reveal that the \emph{GLRT is robust to inter-AP interference, and DTDD doubles the $90\%$-likely sum UL-DL SE compared to traditional TDD-based CF-mMIMO ISAC systems}; while using HD hardware.