You've asked a large number of questions which reveal why it's a bad idea to represent yourself in most court cases, and a particularly bad idea to do so in something as emotional, and as complicated as a child custody case.
A. There's a process called "discovery", in which each side in a civil case can be required to disclose certain information (including evidence, documents, etc.) to the other side, under oath. Both sides can use it. If one party refuses to provide the required information, the court can order them to do so.
B. You're acting as your own attorney. The judge will expect you to be prepare, and to know the law applicable to your case, BEFORE you start trial. If you don't know it, you're not ready for trial, and if you start trial anyhow, the fact that the other attorney knows it better than you do will generally NOT get you a continuance.
C. Who'[s got the burden of proof, at what stage of the proceeding, and what facts you have to prove to meet which burden of proof, is a complicated question, and, per B. above, the judge is unlikely to help you out.
D. If the other party doesn't obey the court's orders, there are various steps you can take to enforce those orders. Any further answer would depend on (1) the order and (2) what the other party did (or didn't do) to disobey the order.
E. The testimony, or statement of preference, of some children is admissible in some circumstances.