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We provide a comprehensive multi-aspect study of the performance of a pipeline used by the LIGO-Virgo Collaboration for estimating parameters of gravitational-wave bursts. We add simulated signals with four different morphologies (sine-Gaussians (SGs), Gaussians, white-noise bursts, and binary black hole signals) to simulated noise samples representing noise of the two Advanced LIGO detectors during their first observing run. We recover them with the BayesWave (BW) pipeline to study its accuracy in sky localization, waveform reconstruction, and estimation of model-independent waveform parameters. BW localizes sources with a level of accuracy comparable for all four morphologies, with the median separation of actual and estimated sky locations ranging from 25°.1 to 30°.3. This is a reasonable accuracy in the two-detector case, and is comparable to accuracies of other localization methods studied previously. As BW reconstructs generic transient signals with SG wavelets, it is unsurprising that BW performs best in reconstructing SG and Gaussian waveforms. The BW accuracy in waveform reconstruction increases steeply with the network signal-to-noise ratio (S/N net ), reaching a 85% and 95% match between the reconstructed and actual waveform below S/N net ≈ 20 and S/N net ≈ 50, respectively, for all morphologies. The BW accuracy in estimating central moments of waveforms is only limited by statistical errors in the frequency domain, and is also affected by systematic errors in the time domain as BW cannot reconstruct low-amplitude parts of signals that are overwhelmed by noise. The figures of merit we introduce can be used in future characterizations of parameter estimation pipelines.