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1. Your father can apply for Indefinite Leave to remain himself without recourse to you. All he needs to do is to fill in the application form. Here is a link to the UK Government website which he can click through to get the correct application form.
https://www.gov.uk/settle-in-the-uk. His eligibility to become a UK settled person stems from you settling and becoming a uK citizen Here is the relevant page https://www.gov.uk/settle-in-the-uk/y/you-re-the-family-member-or-partner-of-a-british-citizen/yes/on-a-work-visa/any-other-work-visa. The form is Form SET(O). Here is a link to the form https://www.gov.uk/government/publications/application-to-settle-in-the-uk-form-seto
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3. Your father is the family member or partner of a British citizen. That is the second link in the pages I directed you to. As you are now a british citizen, he is the family member of you. This applies even if your father previously was the family member of an EEA national. He now applies in a different category as his situation has changed.
4. You have got the wrong end of the stick here. Your father is not obtaining settlement in the UK based on his relationship with you. He has lived in the UK for five years and so has a free-standing and independent right (of you) to settle in the UK. So there is no requirement to leave the UK.
5. He is not an adult dependent relative. That is a totally different category which has nothing got to do with him.