GUNS are placed into various categories, with different levels of permissions required to use or own them. Category D is the lowest level and would correspond to certain air rifles, however he may be applying for D1, which is for other single shot rifles for hunting or sporting use. As you say this category is subject to the weapon being ‘registered’ at the prefecture.
For category C, which also often relates to hunting rifles, a déclaration to the prefecture is required, as opposed to the autorisation required for more powerful category B ones, that are sometimes used for sport.
It is surprising that your father is being asked for his casier (criminal record) details in this case, as usually this is only required for arms subject to déclaration or autorisation, not simple enregistrement. The kind of extract required in these cases is called ‘bulletin no.2’, which shows most serious crimes a person has been convicted of, with some exceptions.
You cannot obtain the French one yourself - the official bodies will do this for you if it is required.
For a copy of a British criminal record statement you can apply to Acro. See: https://www.acro.police.uk/police_certificates.aspx