English: Sentence simplification aims to improve readability and understandability, based on several operations such as splitting, deletion, and paraphrasing. However, a valid simplified sentence should also be logically entailed by its input sentence. In this work, we first present a strong pointer-copy mechanism based sequence-to-sequence sentence simplification model, and then improve its entailment and paraphrasing capabilities via multi-task learning with related auxiliary tasks of entailment and paraphrase generation. Moreover, we propose a novel 'multi-level' layered soft sharing approach where each auxiliary task shares different (higher versus lower) level layers of the sentence simplification model, depending on the task's semantic versus lexico-syntactic nature. We also introduce a novel multi-armed bandit based training approach that dynamically learns how to effectively switch across tasks during multi-task learning. Experiments on multiple popular datasets demonstrate that our model outperforms competitive simplification systems in SARI and FKGL automatic metrics, and human evaluation. Further, we present several ablation analyses on alternative layer sharing methods, soft versus hard sharing, dynamic multi-armed bandit sampling approaches, and our model's learned entailment and paraphrasing skills.