Clearly she has had some advice on how to get as much as she can from you and what to say to the court. They will lie to the court especially when there is no evidence to the contrary. Document everything. Communicate via email and text to get the conversations documented as much as you can.
I would also respond to her custody proposal and add something like the following:
"It unreasonable for you to offer so little overnights, especially after I put them to bed every other night. With that in mind it is only fair, to me and the children, to have 50/50 custody and equal overnights. It is what is normal for them"
This does 2 things. She may contradict herself in a reply in regards to the every other night issue by acknowledging it which you can show the court at the proper time. And it gives credible reason for 50/50 custody.
We sent over our own request for discovery - which is basically equivalent to testifying in court. Questions like - how often did qt put them to sleep? Isn't it true every other night? Did he not take them to school every day? Pick them up and take them to the doctors and activities all the time? Who actually did the housework?
If she answers truthfully the real pictures comes out - I worked my a** off, but I was far from a workaholic guy who comes home puts his feet on the couch and asks his family to serve him. I was involved as a guy working 60 hr a week could be, in addition to paying all the bills. She could lie throughout the whole thing, though. Who knows - she's obviously desperate. She's driving Uber part time I hear from the kids.
Me-45, W-37, T-10 yrs, M-9 yrs D -7 yrs, S-5 yrs BD-5/3/16, D filed 6/8/16