In latin, æ or ae was a (di)graph for the /ae/ diphthong. In English, it is typically pronounced using the /i/ phoneme, which isn't a diphthong. I have heard it pronounced as /ai/ dipthong, but only rarely.

In English, I'm pretty sure the ae digraph is always pronounced as /i/, so calling it a dipthong would probably confuse most people.

Agreed. That is a very good point about it possibly being just a digraph and not always a diphthong. This depends, however, on the use, since it does sometimes slide and other times is just a ligature. The issues seem to stem from the pronunciation changes and the great vowel shift. As you point out, language is a messy business.

A ligature is a character composed of two or more graphemes, it is not a letter

You're correct about that, sort of but not in the sense of language - only for computer definitions. A ligature in language is "a printed or written character (such as æ or ff) consisting of two or more letters or characters joined together" [1]

[1] merriam-webster.com/dictionary/ligature

.... so in other words, it's not a letter in the modern English alphabet. Seems pretty accurate to me.

And hence, not a letter.