Is a woman more likely to have a child born with birth defects if she's had multiple miscarriages, as opposed to a same aged woman that's had none?

No: She may be having miscarriages due to other problems not related to the fetus. Best of luck. Consider seeing a high risk OBGYN.

Depends: We studied a woman that had 17 miscarriages by the time she was referred to the genetic service. A chromosome study revealed an extremely rare balanced defect that would lead to more in the future. However this is the rare exception, you will only gain a good picture of your personal risk if you meet with a medical geneticist and s/he explores both sides of the family.