Sounds good enough, but don’t forget, only commoners call them ablution traps

That was a story, like Homosuck. So it doesn’t happen in the same universe.

She flew there

That was one of John’s last conversations he had with Karkat. John had talked to him several times before so he probably picked it up along the way.

They presumably have some means of education called “SCHOOLFED”. The word is used a few times throughout Act 5 in reference to learning.

They weren’t??

The only direct mention of Problem Sleuth is when Jane talks about her Problem Sleuth poster, when John talks about his Problem Sleuth poster and when he uses his Problem Sleuth game to make stuff

Completely wrong actually. Grimdark deals with the power and influences of the Horrorterrors and their Eldritch Magic.

Problem Sleuth did not take place in the same universe as Homestuck

No

She would timetravel to instigate events (just like her pre-scratch self) she never directly interacted with the other ancestors, except for Condy. But she probably meddled in their affairs