Tell him to look at the vast majority of cooker circuits in the U.K.
Show him this thread and countless others from people whose "electricians" did not know what they were doing.
Yes you do.
New means new - not there before.
There is nothing saying you cannot have a 'new' (there you go; I knew you knew) connection for a hob without replacing a consumer unit.
Your assumption is correct.
It is a repair which is not notifiable.
4mm² would likely have been adequate if the route does not involve running through thermal insulation.
That will be fine.
It is not notifiable. Part P merely states that you shall ensure it is safe.
This is Part P of the...
https://www.legislation.gov.uk/uksi/2010/2214/regulation/12
That is the actual law (in England) and that is all it says.
So, what do you think?
It is not (b) or (c) so is it a new circuit?
You may decide just as well as anyone else.