My child has diphtheria, can I also catch it?

Yes. Diphtheria is highly contagious and possibly deadly disease. You and anyone else exposed to your sick daughter need a course of antibiotics such as Erythromycin and a diphtheria vaccine, regardless of previous vaccination history.
Really?! Diphtheria? Diphtheria is very rare in the United States and europe, where health officials have been immunizing children against it for decades. It does still occur in third world countries where people are not immunized. I see you are from palo alto - not exactly third world. Was your child immunized? If not then hopefully you were. If not then it is very contagious.
Yes & no. If you are fully immunized you have some protection from the disease, yet you may acquire the germ and have a sore throat. You may also spread it to others if you are a carrier. Patients who survived diptheria in the past are not considered immune.