Yeah, can confirm there's problems with this on mobile - I saw the
search suggestions on my phone were there, but they were covered up by
the on-screen keyboard.

Not really sure what to do about it, or how it varies by type of
phone/browser (mine was firefox on a rather small android). Any ideas?

We could use a user agent? For example most iPhones have screens. If enough to show search results. But we could also use @media rules to specify how big the screen must be for results to be shown?

Aye, we could just get rid of them entirely if they're not going to work.

More information on how different phones even behave to begin with would
also be useful.

Do we have a page somewhere describing what all we need to support well?
(or what 'well' even is, or what's good enough)

Nope there's no page describing that stuff. But the results should work as the mobile extension does it unless it also uses a regex based solution on which os and phone your using.

Probably. But that's as good a place to start as any.

Search suggestions cover up the search in ancient safari on ancient iphone.

There is no reason for the rest of the chrome (sitename, dropdown toggles, etc) to remain visible above it while this happens, though (it basically centres the input box?).