Relative to your daughter's lying, she can be disciplined by an automatic time out for any lying of which you become aware. However, you would be prudent to pursue the matter of why she is lying, particulalrly of the sort you describe. The behavior indicates that she feels vulnerable, and this needs to be understood.
There is really nothing to do at her age about enuresis during the night. At her age about 5-10% of children still are wet at night. It's important to be patient and supportive and not display frustration about this.
Re: your last question, there is little you can do - you cannot control your ex-husband's behavior. The children are in a terrible bind. All you can do is take the high road and be the best parents you can be while the children are with you.
There can be any number of reasons why a child might feel vulnerable. Talking with her about her allegation of abuse needn't be construed as a problem. The tone of your voice, your demeanor will set the stage. You might consider having her meet with a pediatric mental health professional and see what guidance you obtain.
Thanks so much for your response, I'm very appreciative.
How can we pursue the matter of why she is lying without making her feeling like she is in trouble? It's hard to get her answer to be more than an "I don't know". I'm lost on why she would feel vulnerable... what reasons do you see most often for vulnerability in a child? I don't know where to begin.