1. Can he continue to work after Jun 29 even his H1B application is not finished processing by USCIS?
-Yes if his H1 is sujected to cap limit (e.g. he works in the Insdustrial company).
2. If he cannot work after Jun 29,
can he stay legally or he has to switch to H4(I am a H1B holder)?
-Yes, not need to change F1 to H4 unless his H1 is denied.
Thank you very much for your reply.