If you admit yourself into a psychiatric hospital can the hospital release you even if you believe you are not well and can you release yourself even if the hospital doesn't believe you're well?

The short answer is yes (even if this is far less than an ideal outcome). People can and do "fake" being well (e.g. reporting lack of suicidal thoughts or lack of hallucinatory activity). In this manner, they technically are stating that they meet the criteria for discharge, and are, therefore, sometimes discharged despite ongoing risk to themselves. More sophisticated therapists and doctors will often recognize that denial symptoms are operating, and will continue to work with at-risk individual, and not accept his/her concrete statements of safety at face value.

If the hospital team doesn't believe you are well enough to be discharged AND you have been involuntarily committed, then you would not be able to release yourself.