Yes, it's does remove search option.
See this article (sorry, in Russian) but s-shot is missing yandex https://vc.ru/services/379414-poiskovik-yandeksa-propal-iz-n...
Grigory Bakunov[1] (ex-Yandex) observes Yandex has been removed (but e.g. OZON is still there): https://t.me/addmeto/4782 (ru, but basically amounts to a screenshot)
Mozilla posts a vague note saying the “default search engine” (NB: omnibox default, not just installation default) may change in Firefox 98, no specific engines are named: https://support.mozilla.org/en-US/kb/search-engine-removal (en)
No, IIRC it just needs to have an appropriate OpenSearch xml file to have support for omnibox. I use a third-party engine that isn't bundled in FF and it works fine in omnibox.