Many websites have headers that are always on-screen when you scroll up, but auto-hide when you scroll down.
I hate them personally because the screen space they would've used up if they stayed permanently on-screen is terrible to use; if you accidentally scroll a pixel up it'll hide a few cm of screen space, and you have to do a little scroll up/down dance to get that screen space back.
Are there any studies on them?
I have no knowledge of any studies, not even on just sticky headers alone but there has been discussions everywhere. It seems like sticky headers became norm but in my opinion as you stated it generates a weird interaction, having the header sticky but hideable and then appearing when scrolling.
I would go with what become a norm, a sticky header but kept as small as possible as seen here on stackexchange and the most websites these days.
Auto-hiding headers are a reactive UX mechanism that some users may consider 'advanced.' Consider having a preference to turn off the feature.
The most important consideration with any possibly 'advanced' UX feature is to follow the same implementation as system-native components and apps. So if the OS native browser implements this UX pattern, look at the published details on its timing, and follow them as exactly as possible.
Among other things, this ensures that even if a given user struggles with the feature, they will not think your app is any worse that then OS itself. Similarly, if the OS updates the reactive curves related to the hiding and showing, be sure to update you app's to follow.
It comes down to execution, its easy to make an annoying sticky menu.
I think there are lots of discussion about the subject but I think these guys have nailed the key points with the best examples.
https://www.clicktale.com/resources/blog/the-3-golden-rules-of-sticky-navigation/
For me a sticky menu is only useful when there is continually actionable item. Like a buy now or something similar. One major benefit is that this pattern is ubiquitous and even poorly designed menus will still be well understood by most users.
