H4/J1 visa for Postdoc and possibility of H4 to J1 conversion

While J1 is preferable due to ease of sponsorship change as compared to H1B, you need EAD if on H4 to accept paid employment. If the primary H1B already have an approved I-140, H4 can apply for EAD.

Doesn’t matter.

No issues.

Yes and no.

J1.