Regarding the instruction of the prosecuting barrister....I could be totally, totally wrong....but I think the CPS do have scope for using barristers in private practice, bot just their own "in house" employees.
I think the idea here is that the CPS barrister already representing the Crown basically told the family that they needed the best criminal prosecutor around, which happened to be Charlotte Rampling. The CPS would then need to instruct and pay her...but she'd still be working for them.
I am no expert...but I think theoretically something like this might be vaguely possible, if highly, highly unlikely.
As far as her visiting them at home, this might just about be possible too, given that Maxine Peake did the same sort of thing when she was prosecuting in Silks!!!!
A proper legal person might tell me I'm talking arse though, which is highly likely.
The exhumation thing is shite....of course the fecking family would be warned first. What, they just show up at the grave and find the coffin being dug up??? No need to have done this for dramatic purposes, it could have been just as dramatic to have them told at home.
I reckon we'll be led to believe that the father meeting up with Ellie's son is sinister and suspicious, but will turn out not to be.
Oh, and Olivia Coleman has to be the best actress working in TV currently. Bloody hell, she's good. The dodgy plot holes are worth putting up with just to watch her, IMO :)