Use new calendar types throughout

With the easy renames out of the way in the last commit, this commit
does the hard work of using the new calendar types:
* Strongly type all method parameters and remove all use of `any`
* Remove now-unnecessary type casts because parameter types are now
  more accurate.
* Ensure all code compiles with strictNullChecks:true and
  strictPropertyInitialization: true
* Remove obsolete types

Author: justingrant