You asked about "claim my son as a dependent". Not sure what the context is. Are you referring to income tax reporting/exemptions? Generally: no.
On the signing school paperwork: Absent a court order granting some form of legal custody or guardianship, step parents have very little rights in regards to their step-children. It's the bio-mom and bio-dad that usually retain legal custody.
You should seek a legal consultation asap.
If she is married to your ex and they file jointly yes. You should go back to court to clarify the issues of doctors visits and psychology meetings. Perhaps if the father is not available to provide transportation it could be necessary especially if you can not attend either. You really need to talk to an attorney
My name is Stephen R. Cohen and have practiced since 1974. I practice in Los Angeles and Orange County, CA. These answers do not create an attorney client relationship. My answers may offend I believe in telling the truth, I use common sense as well as the law. Other state's laws may differ.. There are a lot of really good attorneys on this site, I will do limited appearances which are preparation of court documents it is , less expensive. However generally I believe an attorney is better than none.