Add doctests and update docs

[abhro]

Checklist

• Appropriate tests were added
  • To fix a failing doctest, an implementation of Base.show() was changed (735ea5e), which is covered by said doctest (b79465c)
• Any code changes were done in a way that does not break public API
• All documentation related to code changes were updated
• The new code follows the contributor guidelines, in particular the SciML Style Guide and COLPRAC.
• Any new documentation only uses public API

Additional context

Main changes:

• Update GitHub repository owner name (jonniedie -> SciML)
• Turn almost every code example into a doctest (this revealed some inconsistencies between presented documentation and observed behavior)

[ChrisRackauckas]

Thanks!