Well, Equius was moving toward the basement entrance to Vriska's hive (aka, spidermom's web) before Karkat trolled him, so that initially had nothing to do with anything except his hope that he'd find all the important parts without having to get near spidermom. Later on, though, who knows!

As for Aradia... hmm. Hmmmm. *pose of deep thinking* I will take that suggestion under advisement and we shall see what happens, I guess!