Hi,
Thank you for the details, it helps a lot.
From the screenshot I am guessing a cache related issue. It looks like as if the stylesheet was missing – that explains it perfectly. The cache plugin may not have picked up the search stylesheet, and might be serving an outdated version of the page or asset cache, where the styling is not yet present.
I have tried to replicate the issue by enabling the search in the header module. I tried all the different browsers I had, refreshed like 300 times, but everything was fine. I also tried a few 3rd party website screenshot services, but the styling is all right there as well: https://i.imgur.com/umY706p.png
The plugin cannot “break” itself like that, it must be a missing asset, which is related to site cache (plugins, CDN, page cache etc..)
I checked the configuration, everything is perfectly fine. Clearing all layers of cache will resolve this problem in the future. Right now it looks all right, the cache was probably cleared automatically in the meantime and picked up everything correctly.