Two-way relaying networks (TWRNs) allow for more bandwidth efficient use of the available spectrum since they allow for simultaneous information exchange between two users with the assistance of an intermediate relay node. However, due to superposition of signals at the relay node, the received signal at the user terminals is affected by \emph{multiple impairments}, i.e., channel gains, timing offsets, and carrier frequency offsets, that need to be jointly estimated and compensated. This paper presents a training-based system model for amplify-and-forward (AF) TWRNs in the presence of multiple impairments and proposes maximum likelihood and differential evolution based algorithms for joint estimation of these impairments. The Cramer-Rao lower bounds (CRLBs) for the joint estimation of multiple impairments are derived. A minimum mean-square error based receiver is then proposed to compensate the effect of multiple impairments and decode each user's signal. Simulation results show that the performance of the proposed estimators is very close to the derived CRLBs at moderate-to-high signal-to-noise-ratios. It is also shown that the bit-error rate performance of the overall AF TWRN is close to a TWRN that is based on assumption of perfect knowledge of the synchronization parameters.