I would like to know if it would be possible to put genres for games like "Adventure", "Co-op" to better find the type of game we want speedrun in the searches

And also, better see what a game is about

This feature used to be on the site, but was completely removed with the site redesign. Although, what games had set for them can still be seen on API v1.

I believe people could set engines, genres, and developers.

And with @Pac, there was a feature to search through all these values. But I believe the search part was removed due to spam?

And I think one of site staff did say that they might add these tags back? But I'm not too sure about that now.

The genre descriptions were pretty useless honestly. Every modern game is an action-adventure game with RPG mechanics and optional stealth mechanics, puzzles, and racing mini-games. None of those words mean anything anymore. Engine was also similarly useless for searching, IMO. Having more ways to filter searches that are based on more tangible things would be good, though (e.g. a "co-op" filter could easily include any game with multiplayer categories, rather than having to be manually tagged by game moderators or whatever).

see what a game is about

If you want to see what a game is about, check Wikipedia, or a review, or a gameplay video, etc. This is a speedrunning leaderboard website, not an exhaustive database of video game information that's already well-documented elsewhere.