The game Im talking about has 3 campaigns, each containing unique levels.

The problem here is that all the levels one creates apply for all sub categories (the different campaigns) as seen in the pictures below. How do I avoid this or work around it? Each campaign should have there own, unique levels.

As far as I know, you cannot do that. Categories belong in levels, not the other way around. You just need to list every level like we do in EBTR: https://www.speedrun.com/ebtr/levels

As you can see, we don't separate by cup (because it's impossible), and all the minigames are included in too. Categories which are impossible/redundant should be marked as such by verifying a run which says that the category is invalid, like so: https://www.speedrun.com/ebtr?h=Poker_Kilimanjaro-0-Star&x=l_ldyjnrpd-rklm07wd

So, assuming 'First Blood' and 'Bring it Down' are in the 'Soldier' campaign, you would name the levels like so:

Soldier - First Blood
Soldier - Bring it Down
Soldier - Third Level Name
Spec Ops - First Level Name
Spec Ops - Second Level Name
Spec Ops - Third Level Name
Zombie - First Level Name
Zombie - Second Level Name
Zombie - Third Level Name

A game I like to speedrun does not have a genre,

what should I do about it? Can I make my own? Could I request one to be made?