Speech enhancement systems, which denoise and dereverberate distorted signals, are usually optimized based on signal reconstruction objectives including the maximum likelihood and minimum mean square error. However, emergent end-to-end neural methods enable to optimize the speech enhancement system with more application-oriented objectives. For example, we can jointly optimize speech enhancement and automatic speech recognition (ASR) only with ASR error minimization criteria. The major contribution of this paper is to investigate how a system optimized based on the ASR objective improves the speech enhancement quality on various signal level metrics in addition to the ASR word error rate (WER) metric. We use a recently developed multichannel end-to-end (ME2E) system, which integrates neural dereverberation, beamforming, and attention-based speech recognition within a single neural network. Additionally, we propose to extend the dereverberation sub network of ME2E by dynamically varying the filter order in linear prediction by using reinforcement learning, and extend the beamforming subnetwork by incorporating the estimation of a speech distortion factor. The experiments reveal how well different signal level metrics correlate with the WER metric, and verify that learning-based speech enhancement can be realized by end-to-end ASR training objectives without using parallel clean and noisy data.