Generally, you should be able to study on a part-time basis but please schedule an appointment with an immigration lawyer, so she/he can evaluate all of the facts and give you advice. Aren't your Phd classes related to yout work?
Yes, provided they continue to maintain their H-1B status.
J Charles Ferrari Eng & Nishimura 213.622.2255 The statement above is general in nature and does not constitute legal advice, as not all the facts are known. You should retain an attorney to review all the facts specific to your case in order to receive advise specific to your case. The statement above does not create an attorney/client relationship. Answers on Avvo can only be general ones, as specific answers would require knowledge of all the facts. As such, they may or may not apply to the question.