Travel to US on H4 while H1 is in-progress

[Sasibhushan]

Could you please clarify the following queries?

Case1: if applicant have valid H4 visa and filed H1B from outside US as normal consular process but not yet approved
a) Can applicant travel to US with H4 without impacting in progress H1B application?
b) Can applicant apply premium processing and change of status (COS) same time after enter into US on H4? What will be the fees?
c) Can applicant should wait until 1st October to work (or) eligible to work as soon as H1B and COS approved?

Case2: if applicant travels to US on H4 and filed H1B as change of status (COS)
a) Can applicant file H1B and COS on same time as premium processing?
b) Can applicant should wait until 1st October to work (or) eligible to work as soon as H1B and COS approved?

Thanks