How often should lithium level, kidney function, thyroid etc. Be checked? Doctor has not requested since monthly test when starting for 3 months

Often. If I read you correctly your doctor should get your lab data once and, if things are OK, there appears to be no need for a study within three months. That said, if you are not comfortable with that doctor I recommend you make an appointment to see a Psychologist - they are excellent in taking care of patients with concerns. Please let me know how things go.