Handling your own divorce pro se, particularly involving discovery disputes, particularly across state lines is not a very good idea. Pro se parties are severly disadvantaged by not knowing the rules of procedure and evidence.
But as to your questions:
(1) He can ask for your medical records. And you can object to producing them and even ask the court for a protective order. Whether you would get such an order, I don't know.
(2) The documents you produce in discovery are not part of the public record anyway. You don't need to worry about sealing records for that reason.
(3) You would ask the opposing attorney for an extension. He should grant you one. BTW: Did you serve discovery on your husband? You should. Even if you just copy theirs and change the names/pronouns/party designations, you should send your own discovery requests.
(4) If the opposing attorney won't give you more time, you can ask the court for more time. Most people can respond to the discovery in a day or two, so maybe you have time. NOTE: when responding to requests for production, you only have to produce what you actually have. You do not have to create documents or, for example, ask the bank to print old statements that you don't have. You only have to produce what is in your possession or control.
You definately need to hire an attorney.
If you need an attorney that offers a payment plan, call Patricia Bushman at 713-807-9405. She usually won't talk to you on the phone. She makes you come into her office. I rent space in her office. I'm not taking cases because both of my parents are both very ill.
The discovery information just goes to the other side and does not go to the court so it's not in the court's record. So don't worry about anyone seeing it.
Please hire an attorney.
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