1) It was one of Dylan's pediatricians who reported Allen to the police:

This is not really meaningful because by law the doctors have to call the cops. Duty to Report it's called. This is even if the doctor does not turn up evidence of abuse. I have a question, can you find a medical professional who thought Dylan Farrow was abused? I haven't come across one.

There were two investigations one in NY and one in CT.

Mia Farrow phoned Dr. Susan Coates, a child psychologist who worked with the family. Coates called New York City Child Welfare. This kicked off the NY investigation. They would investigate for 14 months and then conclude no abuse occur.

New York State child welfare investigators have dropped their inquiry into the charge that Woody Allen sexually molested his 7-year-old daughter, saying they consider the accusation unfounded.

After Farrow called Coates, she called a lawyer, the lawyer suggested to take her to a pediatrician. They go to a CT Doctor and Mia later testified that when asked Dylan said her father touched her shoulders. I don't think this resulted in the police being called because Farrow would go to another doctor a few days later. (I'm wrong. Vanity Fair says the police were called on Aug 6th) But before that doctor visit, over at least the next day Mia Farrow makes a video where she interviews Dylan. The video is not continuous. It stops and starts. A nanny would say in a deposition that this was made over 2 or 3 days. So you do not see what Mia and Dylan say off camera. I just learned today that Linda Fairstein head of the Sex Crimes Unit in NY for 25 years said most prosecutors were put off by the video and disturbingly Dylan was naked for part of it. The Yale team believed coaching of Dylan occurred when the camera was off based on the fact that what Dylan's story was not consistent. http://video.foxbusiness.com/v/3151187744001/?#sp=show-clips

They go back to a different pediatrician with the video four days after the original incident

2) The panel of abuse experts were negligent in handling their records, and they all refused to testify under oath in court about their report (possibly because they didn't want a public record available in the court records about their negligence?)

This is complete spin. This was a very well respected team dealing with Child Abuse. See the Linda Fairstein interview above. They were not negligent. It was their standard procedure that once they finish their report, they got rid of their notes. The report is the work product that stands not their raw notes. The FBI does something similar. They offered a sworn statement in the custody case. The first couple of pages of the report are available here.

3) While the prosecutor declined to charge Allen criminally

Um, no. The CT prosecutor could not bring a case because the official State Police investigation stated flatly. "It is our expert opinion that Dylan was not sexually abused by Mr. Allen."

And what is your basis for this claim? (because Allen's lawyers made it clear they would torment Dylan on the stand if the case went forward) The XO Jane link you post certainly doesn't provide any basis and it barely has anything to do with the facts of the Allen case, as the author never mentions any.

There was a family court case...

Exactly. Woody Allen was never charged with assaulting Dylan Farrow. He was never found "not guilty" because there was no criminal case in CT or in NY. There was a custody case and Farrow won custody.

So, yeah, Woody Allen's lawyers gamed the system to keep him from being charged criminally,

Complete Bullshit. Give a single example of this.

How did he get the CT State police investigation to conclude no abuse occurred?

How did he get the NYC Child Welfare investigation to say the charges were unfounded.