I think you should get an attorney for this hearing. Even though the courts have a preference for the status quo, you can never be sure how any individual court may rule. Your mom can be your witness. And you should gather your records regarding child support payments showing that he is in arrears. That is very good evidence of his motives. Good luck.
Stop spinning your wheels. Most likely the emails are being written with your ex's direction and approval. She may just want the clarity it provides and she Appreciates helpful input. She is doing nothing wrong. Asking as the best interests if the children are being served then there is no harm.
If the other party is self represented then your attorney can speak to him or her. However your attorney cannot advocate for the other party He may be trying to settle your case which would be in your best interests
Yes this can be admissible but you should talk to an attorney about how to do it correctly. In the mean time you should take screen shot to preserve the evidence. The photos need to be date stamped and the person who took them will have to come to court to verify them. Again an attorney can help you to do this correctly.
If he does not show up, and he probably will not since he he working outside the US, the court will
probably issue a bench warrant for his arrest and continue the hearing. If he shows up, it is your burden to prove that there is a valid order for support and that he had knowledge of the Order. He then has the burden to prove that he did make the payments. But you can't get blood from a stone, so do not expect any payment soon.