docs: add unified search & styling

### What feature or improvement would you like to see?

Noticed some unfinished tasks in https://github.com/apache/arrow-adbc/issues/1357

- With [pagefind](https://pagefind.app/) we could let you search the Doxygen and Javadoc sites at the same time as the Sphinx site
- We probably want some way to allow searching them separately so that different language results don't cross-pollute too much?
- An example here: https://blog.gandi.net/en/posts/sphinx-doc-and-pagefind/
- A [custom stylesheet for Javadoc](https://docs.oracle.com/en/java/javase/21/javadoc/javadoc-css-themes.html) and for Doxygen could help the different docs look more cohesive?

Start by reviewing the unfinished tasks in issue #1357 and the linked Pagefind and Sphinx example. Inspect how the Doxygen, Javadoc, and Sphinx sites are currently published; done means unified search is available with a way to separate language results, and the generated sites have a more cohesive style.

Written by the indexing model from the issue text.