Confidence scores are very useful for downstream applications of automatic speech recognition (ASR) systems. Recent works have proposed using neural networks to learn word or utterance confidence scores for end-to-end ASR. In those studies, word confidence by itself does not model deletions, and utterance confidence does not take advantage of word-level training signals. This paper proposes to jointly learn word confidence, word deletion, and utterance confidence. Empirical results show that multi-task learning with all three objectives improves confidence metrics (NCE, AUC, RMSE) without the need for increasing the model size of the confidence estimation module. Using the utterance-level confidence for rescoring also decreases the word error rates on Googles Voice Search and Long-tail Maps datasets by 3-5% relative, without needing a dedicated neural rescorer.