You can ask to sign out AMA (against medical advice) and then they can decide if they want to take you to court to be involuntarily committed. At the hospital I go to (been there many times) they have 12 hours after you sign AMA before they have to let you go. That gives the doctor time to get in to talk to you. I did it once and after talking to the doctor I decided to stay voluntarily.

As to your desire for in depth testing...The ER is not the place for that. It is definitely for Emergency treatment. They don't do tests for people coming in for mental illness problems. Their role is to find an inpatient bed for you or determine that you don't need to be admitted. Once you are on the psych ward, their roles are to keep you safe, and get you stable, or at least not actively suicidal or homicidal or psychotic, with medication.

What should you do to get out? Follow the instructions of the staff and take your medications. Participate in any groups offered. Clearly explain your situation of having psychosis but safe to the psychiatrist. As to having neurological testing done, that is rare in a psych ward. You could ask the psychiatrist but I would be surprised if it is done as insurance only pays if there are symptoms, such as half your face drooping, that justify having tests done.

Hospitalization doesn't have to be a bad experience. When I am admitted and the ward door locks behind me I feel immediate relief as I know that I don't have to fight the desire to suicide as much. I understand that because their job is to create a safe environment I won't be able to have my shoe laces or a spiral bound journal. I am willing to give up some things to be safe from myself and my actions. Hospitalization can be a good thing and the right thing to do.