H-1s are site specific.
Your employer needs to retain an experienced immigration lawyer to review all the facts, advise them, and handle the case. Your employer can find one through http://www.ailalawyer.com.
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.
Until you are granted H1b status, you can do whatever you could do before on your L1.
The above is intended only as general information, and does not constitute legal advice. You must speak with an attorney to discuss your individual case.
Probably No. Check with your immigration lawyer for the best advice.
Attorney: Immigration Law
13295 Illinois St Ste 128
Carmel IN 46032
Ph: (317) 660-6174
Contact (317) 660-6174 for specific legal advice. Answers here are not legal advice because they are of general nature and not tailored to your specific situation. You should not act on this answer without checking with an Immigration Attorney.