Numbered as I can't be ar*sed to separate them
1. They are Quasi-Judicial processes. Similar perhaps to employee disciplinary processes. Things have to be transparent, fair and impartial. Nobody should be having chit chats about what was really meant by the report. Comments about the leniency of the Police fine etc, do not aid impartial decisions about penalties and sanctions. if you found yourself in a fact finding interview and the report was mild, but there was an informal discussion about what was not said for example, you would rightly cry unfair.
2 I don't either, but it's probably not now going to be seen as impartial given the job offer. You don't accept a job from the Labour Party unless you are passionate about the cause. It would be unlikely that JohnD would be offered or could accept CoS for Sunak. Sue clearly is and was a Labour person as is JohnD.
3. As previously said, nobody can stop her. I suspect though Sir K, didn't quite expect the problem she has caused even before she starts working. Perhaps he smelt the juicy prize and didn't think about the consequences to both their reputations. She almost certainly knows where the bodies are buried having worked as a senior civil servant for a very long time. As Ackland-Hood said [paraphrase] it's not possible for all this to have happened without a significant prior contact, which was not reported = Rules broken.
www.gov.uk
On the bases of her fellow civil servants commentary and shock, its fairly clear that Integral Sue did break the code. Naughty Sue.