I have a game that I think can have several categories but I don't think many people have herd of it

That doesn't matter. There are many games on the site with only one runner, so game popularity is rarely a factor.

As soon as your account is at least 7 days old, and your game complies with the game request rules, you can request it for the site. Note that even if the game is rejected for some reason, nothing prevents you from speedrunning it anyway.

The game I want to run has no active mods. How would my run get verified? Would I have to become a mod? If so, how? If not, then how would the run get verified?

Go to https://www.speedrun.com/The_Site/thread/dueac if you wish to request for Moderator status for a game with inactive mods. However, attempt to get contact with any of the mods through other means before requesting Moderator.