I believe the LeelaChess bot on Lichess is hosted by
lczero.org, and they are using their own hardware, yes.
The difficulty settings are something that Lichess has only. I am not sure how it works exactly, but you could think of it as this: "level 1" gives 0.5 seconds of thinking time to Stockfish, "level 2" gives 1.0 seconds, and so on, until "level 8" gives 5.0 seconds of thinking time. So the higher levels will be harder since Stockfish thinks for longer.
But for every level, Stockfish is still running its latest version (which is version 9, not to be confused with the "levels" here). It's simply configured to think for different times, and thus allow for different difficulty settings. I've maybe complicated the explanation, but that is how it works.
And in the video you've posted, they're playing Stockfish version 8 against Stockfish version 9 at full strength on their own hardware. You wouldn't be able to know what Lichess "level" that their SF v9 is running at, without first knowing the kind of CPU they have and so on.