You'll have to wait seven days to create the leaderboard, then. When you request it, make sure it's under its proper title, "Lawyer Guy: Defender of Justice".
You'd have to purchase a capture card for the system. I don't know much about those for the DS, but they might be a little difficult to come by.
It looks like this question has already been answered on the forums. This post suggests that the PC version of Doom 64 (assuming you're not emulating) was released in 2020, in which case runs would go on the Doom 64 (2020) leaderboard.
You can request a game to be added to this site, but be aware that webgames aren't always sufficient quality to be accepted.
For the second question, you'll want to be using the program LiveSplit. This is a timer which you can manually control, although this is almost always only for the runner's benefit and not for moderators. You can use a window capture in OBS Studio to place it wherever you want in your video as you record.
Did you look at your resolution, framerate, and bitrate settings?
Only very occasionally these days, I'm afraid. There's plenty of more engaging speedrunners out there, you should check the Streams page or SpeedRunsLive for more.
There's many screen recorder programs on the internet, the most popular of which is OBS Studio (which is free). I'd recommend looking into how to set this up so you can get times up on a leaderboard.
It's worth looking at the Bad Games Done Quick blocks of prior GDQs, they aren't easily searchable but you can just go through the recordings.
I'm no expert, so some of my analysis may be incorrect, but there's some strange vertical lines next to some of the digits in the image, as if someone made some sloppy edits. Speaking of which, the image is pretty clean, suggesting emulation. And the best laps aren't red like the final times. But we're forgetting the most important rule...
no vid, no did
P.S.: @Avriix your YouTube link is broken
Correction to @Timmiluvs 's Nintendo claim: GameCube games were playable on original Wiis, and the Dolphin emulator successfully emulates both. Those are 3D systems, however.
Your best bet is something like BizHawk or RetroArch, though I think BizHawk focuses on having more accurate cores.
Did a quick Google search for "Nestopia 1.40 recording" and found that a few users reported broken recordings on certain games, it's possible this is happening here. Not sure if there's anything you can do within Nestopia to help, but you can always use an external program (OBS) to record or just use a different emulator.
I'd guess when the intro ends, looking at some longer runs on this leaderboard, but I'm not sure.
Your question is confusing, can you elaborate? If you mean "display what your IGT is for the game you're playing", that's a thing that varies by game. You could search how to do that elsewhere, or just tell use what game you're speedrunning.
Recording a video and uploading a video are two different processes. For recording, you'll want to look into using software such as OBS Studio, which can record your screen and give you a video file. You can then upload this video to YouTube, Vimeo, or other similar websites and copy that URL and submit a run on speedrun.com.
https://www.speedrun.com/Bourasque/allposts Not much to go off of at the moment but certainly suspicious.