I am a Gnus user. My 2 cents…

concerning (1): I have no offline sync in place, all my emails stay on the server. The IMAP protocol has a decent server-side search included[0], combined with Gnus unified search syntax[1], I enjoy a hassle-free search experience.

[0]: https://www.gnu.org/software/emacs/manual/html_node/gnus/Sea...

[1]: https://www.gnu.org/software/emacs/manual/html_node/gnus/Sea...