I just played through this game for the first time yesterday, without looking up what anyone else was doing for any particular category. My interpretation of "Beat the High Score" was to score over 10,000 points in Tetris. Not sure if that's a goal any runners are interested, but it made sense to me in the context of the game so figured I'd throw it out there as an option that I haven't seen mentioned.

The practice hack is an IPS patch that needs to be applied to the original game. I use LunarIPS, but I think there are a couple other options out there.

The leaderboard defaults to NES. ACE can only be done on FDS. If you switch to FDS, you'll see all the verified runs.

Pretty sure NES Mini is an acceptable hardware. In that case, you could record your gameplay exactly as you did that video.