Landing page improvements

### 📝 Description of the feature

List of landing page improvements:

- Use sphinx-autoapi.
- Unify documentation sections.
- Cleanup of project's root directory.

### 💡 Steps for implementing the feature

_No response_

### 🔗 Useful links and references

_No response_

Start by reviewing the landing page and the project's current documentation sections. Investigate how sphinx-autoapi would fit, identify what documentation should be unified, and determine which root-directory cleanup items are in scope; done means these three areas have a clearly defined, consistent result.

Written by the indexing model from the issue text.