I'd prefer a reputable courier company over the post office. That said, my experiences with FedEx for items from overseas have not been uniformly good, but the problems have been with confusion (theirs, not mine) over payment of customs duty and the like, not with actual delivery.

Use of the post office depends in part on the postal service at the other end, so a lot must depend on what country your family member is in. If she's in the Netherlands or another developed country with good postal services, it's obviously safer than if she's in a country with poor or erratic postal services.

However you send the passport, before it goes away I'd take a couple of copies of the photograph page - keep one copy yourself and get another copy to her by separate cover (e.g. by fax or scanned in by e-mail).

I don't think she broke the law in leaving the US on her non-US passport. US citizens are required to present themselves to US government employees as US citizens, but check-in agents and other airline staff, to whom she presumably had to show her passport, are not US government employees. Anyway, that's hardly a problem now.