回复:H1B transfer & 父母探亲签证,紧急求助!
1. Your school laywer is right. Your dependents are legal as long as both their H4s and your H1 are valid.
2. I do not think that the sponshor name is an issue. If you worry about it, file a 539 for them to get a new I-94s. However, the processing may takes a few months.