W Venn - We cannot provide guidance on how the executor should proceed in selling the house as that would be going beyond our remit. You need to speak to your legal advisor on the matter. We can only provide guidance from land registration perspective.
The first issue to consider is what is the intention of those involved - is the title to be registered in the name(s) of the beneficiaries of your mother's estate or is it to be sold straight away.
Guidance on our website on GOV.UK under
How to update property records when someone has died may be the best place to begin with.
With regards to your father's name still being on the register, as he passed away before your mother, an official or certified copy of the Death Certificate would suffice our requirements. An official or certified copy of Probate for your mother's estate is what we would be looking for mainly. These would have to be accompanied by the relevant land registration forms and fees.
On the supplementary question, that is something you need to discuss with your legal advisor.