ENH: EADtype._find_compatible_dtype
- [x] closes #24246
- [x] closes #45349
- [x] closes #32346
- [x] [Tests added and passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#writing-tests) if fixing a bug or adding a new feature
- [x] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit).
- [x] Added [type annotations](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#type-hints) to new arguments/methods/functions.
- [ ] Added an entry in the latest `doc/source/whatsnew/vX.X.X.rst` file if fixing a bug or adding a new feature.

Tests copied from #52833, so this may close #52235.

Needs a lot of cleanup, in particular docs.  The ArrowDtype._find_compatible_dtype implementation is quite kludgy, would appreciate help from someone more knowledgeable.

I think `_find_compatible_dtype` is the "correct" long-term non-kludge solution for setitem-with-expansion.

This pull request is stale because it has been open for thirty days with no activity. Please [update](https://pandas.pydata.org/pandas-docs/stable/development/contributing.html#updating-your-pull-request) and respond to this comment if you're still interested in working on this. Updated with some added tests, changed EADtype._maybe_promote->_find_compatible_dtype.

There are some TODOs mostly for pyarrow and related to nan-vs-NA, but I don't think any of them are blockers. _find_compatible_dtype can be used to implement a default implementation of _is_valid_scalar_for, which can be used to implement _from_scalars (both discussed on this week's dev call).