In terms of intimate conversation, should I ask her whether she's still seeing the OM? Or how she's handling the people around town who are probably learning of the A? These will be tough questions for her to hear but I doubt anyone else is asking her about these events.
Stay away for any intimate conversations! The outcome would be bad. Do not ask her any questions about her personal life, OM, or the people around town.
Leave the conversation up to her. It is not up to you to try to entertain her when she's with you. Keep it light & easy. Talk about the weather or the kids.
If she doesn't want your cell number when you are over seas....that a big sign that she hasn't any intentions of contacting you. However, you need somebody to be able to reach you for emergency reasons regarding the children.
It is not about what you feel should work in your M. It is about doing the work that gets the right results. Do what works!