During the Stridereunion, Dave said that his abuse at Bro's hands made him dislike seeing blood, being in danger and hearing metal noises.

He got a planet full of blood-like lava, covered in gears which probably make metal noises nonstop, and which probably is the most outright dangerous out of the Beta kids' planets, as there is a constant risk of falling into the lava or being crushed by gears.

Karkat loathed his own mutant blood, as he knew it would be his death if discovered, and his greatest fear was that someone would find out about it.

He got a planet completely covered in his own blood and full of sharp rocks which might cut him (and reveal his secret) at any moment.

We don't really know about Latula's planet, but we know that Aranea talks a lot about how she overcame her lack of a sense of smell in order to avoid being culled.

That makes me think that Knight quests have to do with overcoming one's fears and insecurities, and that their planets are based on things they fear and hate in order to challenge them.