Sounds like your lawyer is starting to set you up for the fall. She will depose when you instruct her to depose. She is your employee that you have contracted with, don't sit around and wait for her to figure something out, direct her! Of course nothing was said as why would she want to tell you about the dude she probably just banged out.
The point we are making is you are making the mistake of thinking this is the same person you met and had children with. She is no longer that person and no longer of your concern. Not only that, stop giving a rip what she thinks and cares about, that is no longer you concern. My STBX about once a week starts railing on about how she needs money blah blah blah, then she goes out drinking or buy's smokes! You need to understand that more than likely, she is lying to you from here on out unless you see it with your own two eyes.
something close to 50/50 isn't 50/50. You need to stop thinking what you thinking your ex will accept and start pushing what YOU want. This is the time to be self centered with the children's best interest at heart.
Oh, and from here on out, if she wants to bang out some dude, she needs to foot the bill for that, no more kicking you out of the family home and frame it as spending time with the kids. When are you ditching that other place? Depending on your ex, your June time frame is going to get blown out of the water. You are setting up status quo that you are paying for her place along with your place, is that what you want to do? Pick a place and get rid of the other one.