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...
gnus had some massive IMAP performance improvements a few years (probably close to a decade now) ago. Before that it was quite painful to use it on large mailboxes without a local imap - I used to sync that with offlineimap. When they had a massive issue moving from python2 to python3, and keeping that running on a modern distro started getting painful I tried it without local imap - and realised those improvements made things fast enough that you can run it on remote mailboxes, and even do so in your main emacs instance.