We study an uplink scenario of a cell-free massive multiple-input multiple-output (CF-mMIMO) system with limited capacity fronthaul links (LC-FHLs) connecting each access point (AP) to the central unit (CU), where user equipments and APs are subject to hardware impairments. Therefore, to efficiently use the capacity of FHLs to maximize the achievable rate, we analyze three strategies for performing compression and forwarding of channel state information (CSI) and data signals over the LC-FHLs to the CU; Compress-forward-estimate (CFE), estimate-compress-forward (ECF), and estimate-multiply-compress-forward (EMCF). For CFE and EMCF achievable rates are derived, and for ECF one upper and lower bounds are presented which are tight for ideal hardware and FHLs. Also for forwarding the quantized version of CSI and data signals of each user, low-complexity fronthaul capacity allocations are proposed for ECF and EMCF strategies, which considerably improve the performance of the system, especially for limited capacity FHLs. Our results indicate that at high SNR regime and for large enough capacity of FHLs, estimating channels at the CU rather than APs result in smaller estimation error. Then, geometric programming power allocations are developed for CFE and ECF to maximize sum rates. Finally, to highlight the performance characteristics of the system numerical results are presented.