[EPIC] API docs redesign phase 2

[Eric-Arellano]

We made huge improvements to API docs in https://github.com/Qiskit/documentation/issues/479. This epic is to track other follow-up improvements we still want to make.

### Tasks
- [ ] https://github.com/Qiskit/documentation/issues/1098
- [ ] https://github.com/Qiskit/documentation/issues/1388
- [ ] https://github.com/Qiskit/documentation/issues/1395
- [ ] https://github.com/Qiskit/documentation/issues/1361