A: No. A qualifying cap-gap filing can extend your F-1 status during the gap, but if your OPT work authorization had already ended when the H-1B petition was filed, there is no employment authorization to extend. You can remain in status under the cap-gap rules if you qualify, but you cannot restart work just because of the cap-gap.
Source: USCIS Policy Manual, Vol. 2, Part F, Ch. 5: https://www.uscis.gov/policy-manual/volume-2-part-f-chapter-5