Which other health problems are caused by whooping cough?

Seizures, apnea. Besides the terrible cough, whooping cough (in young babies) can cause cessation of breathing (apnea), seizures, and death.
In unvaccinated... In unvaccinated children, death and brain hemorrhage are likely possibilities (they cough so hard, they can exhaust their breathing muscles or burst a blood vessel in the brain). Even without complications, constant cough for 3 months is a highly unpleasant experience.