Multiple bugs in `multiprocessing.managers`: pickling, nesting, etc

Bug report

Bug description:

I noticed multiple bugs in multiprocessing.mangers and explained them, with fixes, in a blog post: https://zpz.github.io/blog/python-mp-manager-2/

A summary of the bugs are as follows:

• Pickling of BaseProxy objects: easily causes object release before unpickling happens in another process
• Unable to call proxy-returning methods on a proxy that has been passed to another process. (There is an open PR https://github.com/python/cpython/pull/99598 but my solution appears simpler and cleaner.)
• "Nested proxies" are never released (also reported in this isseu https://github.com/python/cpython/issues/103136)
• A simple enhancement (not exactly a bug): server-side exceptions lose all traceback info when propagated to user (also reported here in this issue https://github.com/python/cpython/issues/78279)

In addition, this issue

• The specification of exposed is a mess. (See https://zpz.github.io/blog/python-mp-manager-2/#exposed-methods)

My package mpservice.multiprocessing.server_process fixes these issues with some additional enhancements and simplifications. The enhancements are covered in another blog post: https://zpz.github.io/blog/python-mp-manager-3/

I hope to engage in some discussions (esp with a core developer) before working on some PR, either local fixes or something a little more holistic.

CPython versions tested on:

3.10, 3.13

Operating systems tested on:

Linux

multiprocessing.managers のエントリーポイントとリンクされたブログ記事から始め、報告された挙動を参照先の PR と issue と比較してください。pickling、ネストされたプロキシ、例外の traceback、公開されたメソッドのケースを再現してください。スコープについて合意し、選択した各バグに回帰テストが用意されて初めて作業完了となります。