vH
Answering your questions, and having no regard for the unfortunate accident in which I suspect none of us know what happened.
There is no other reason to carry passengers on aerobatic flights
Aerobatics, correctly and safely conducted as taught, are no more hazardous than normal flying. If the aircraft remains in W&B, then whether passengers are carried or not is irrelevant IMHO. Only concern I have is that the passenger is keen, briefed and is checked throughout to ensure they are happy to continue.
Do Aerobatic pilots do HASELL checks or are they "beyond" these?
Yes. As per stalling, adjusted (e.g. Min Hts) as required. 'Security' becomes particularly important.
What is the standing rule for carrying passengers when executing aerobatics?
See above.
What is the minimum acceptable height for aerobatics when having passengers on board?
Depends on type and pilot experience / training. In general, same height as without Passengers,
except when getting into "display" aerobatics and practices, when heights drop and passengers are less suitable / prohibited. As a guide (but others will have different opinions) 3000' might be a good starting point as a "no go below". Adjust that as other factors see fit...
NoD
PS The USA has a strange rule about passengers requiring parachutes for aerobatics. In itself, it is meaningless without adding training, whether you can abandon that aircraft type, min heights for spin abandonment etc.