Hi ! So today I beat the WR of a game (Another Code Two Memories) that I am moderating but the leaderboard refreshed in a strange way: I am therefore 1st but also 3rd, as before: my two runs were both displayed on the leaderboard. To correct this, I deleted my old run: it fixed the problem, but it created another one. My old run was the WR before, so by removing it, part of the WR History is now erased. So, is it possible to put a deleted run back in the WR History? I think no and it doesn't matter at all, but if it's possible it's still nice ;)

Did you run on a different platform/region for your second run? When one of those options are disabled, the leaderboard will show both runs: 
Regions obsolete each other [?]
Platforms obsolete each other [?]

1. Don't delete runs unless you really want them deleted; it won't solve any problem, and as you said, it might change the WR history.
2. The two runs showed together on the leaderboard, probably means it was set with "Platforms obsolete each other" or "Regions obsolete each other" set to NO in the settings.
In that case, two runs by the same runner but on different regions (PAL/NTSC/JAPAN) will both show on the board.
3. You can submit your deleted run once again, it doesn't matter that you have a faster run on the leaderboard. Just set the date to the correct date you did the run, instead of today date.