I just wrote the following paragraph as an edit to this answer:
Using LH2 for regular burns to maintain ISS altitude would require a whole new set of engines that may not yet exist anywhere. While LH2 is one of the main workhorses getting big things into space from a launchpad (e.g. Saturn V except 1st stage, Space Shuttle) it's not usually kept stored for long periods, and engines designed for long term usage and multiple starts over time without maintenance would be a technical challenge. Engines that use cryogenic fuels are substantially more complex to build, operate, and maintain, as they have components that require pre-ignition cool downs.
So you can see that I have a hunch that the answers for "Have reliable, restartable LH2/LOX ever existed? - ever used?" are probably "No" and "No", that LH2/LOX is only fueled at launch time and ignited once, but I really don't know.
Has there ever been an effort to build or test a restartable LH2/LOX engine? Even one restart? Here restart means in space, not in a test rig or launch pad on Earth.