In Flarum's bundled Emoji extension, the "auto-complete" and "picker" are essentially the same feature. Removing either would remove the other.
I suppose it could be a setting, but it would actually be much easier to just create a separate extension that replaces native Unicode emojis with images, as it's the only other feature of the bundled Emoji extension.
Using Flarum without the Emoji extension will not prevent users from inserting emojis in their posts using the operating system native input. The emojis will just be kept as native Unicode points and will look different depending on the device used to read the forum.
EDIT: the "weird behavior" is the currently the expected behavior due to us re-using an emoji list and showing it in the same order. The list used to start with smileys, but the third-party maintainers added symbols at the beginning of the list in an update. We should definitely change it and a PR for it would be welcome, but it's currently not at the top of the priority list. There's a discussion tracking this proposal here https://discuss.flarum.org/d/31202-show-more-or-different-emojis-when-typing