It turns out that Thunderbird can actually do all these things, but it requires tweaking variables using a tool like about:config in Firefox. In Thunderbird, this is accessible by going to Edit/Preferences/Advanced/General/Config Editor have no idea why (some of) these options aren't user visible outside of the config tool, but here they are:
mail.server.default.using_subscription should be false to get Thunderbird to look for new folders.
mail.check_all_imap_folders_for_new shuld be true to get Thunderbird to check all those folders for new messages.
res should be true to get Thunderbird to download the new messages before they are accessed. This may need to be created (right click in the table area and select New/Boolean)
mail.server.default.download_on_biff should be true. I think this helps out with autosync (may be redundant).
mail.strict_threading should be true to get Thunderbird to stop outsmarting itself with respect to subject-based email threads.
I think that covers it. I'd also like to get my google calendar synced into Thunderbird and again synced into my GNOME desktop calendar like Evolution does (there are plugins for that), but this should be enough to satisfy my critical email needs in Thunderbird.