As a native cantonese speaker, I understand what you were trying to ask. Basically, 啲 isnt a common word in written context. There are two main usages of this word When the sound is di1, the letter "d" and "啲" can be used interchangeably. For example, are you feeling better? 你好d未呀？ or 你好啲未呀？ Ok when the sound is dak1, actually it should written as "的" formally. It just means "of" here.

I am not sure about this one. I have never heard people saying "an1" for "奀".

Another example is 奀 : Actually it's ngan1 but in spoken it becomes an1

I am not quite sure about your question. As far as I know, "啲" is "di1", "的" is "dik1". They have the same meaning, but "啲" is spoken Cantonese.