1) Motion and petition are similar, but a petition MUST be in writing, and generally if not answered to within 21 days (Or otherwise by court order) can be advanced by filing a motion for default, which has the effect of making the pleadings judicial admissions.
2) Hearsay, google it. I can't cross examine a letter. I need to subpeona the doctor, almost certainly disclose the identity of this witness to everyone in advance, and the doctor must say something that the court feels is relevant.
3) Get a lawyer. Call legal aid. Something. Anything.
A petition is like a complaint in a lawsuit. It lays out facts that happened outside court and asks for the judge to intervene and award relief such as a monetary award, custody of a child or a divorce. A motion generally seeks lesser relief and it cannot start a case off.
You appear to be engaged in a matter of significant importance to you and your child. This is no place to pretend to be a lawyer as major harm can come to you, your child or your relationship with that child. Get competent legal advice. Get a lawyer NOW.
Please be sure to mark the best answer to your question. My answers are general and do not form an attorney-client relationship. I'm happy to talk to prospective clients in my areas of concentration and geographical location.
A motion is a written or oral application to a court in a pending case seeking some sort of ruling or order. A petition, on the other hand, is always in writing, and is considered a pleading, used to commence a proceeding, or initiate a collateral one. I strongly encourage you to seek out and retain an attorney. Good luck.