I think עוֹרֵך דִין may also be translated as solicitor (Britain) or attorney (US)
The word עורך on it's own means 'editor', but as a verb it can mean something like getting something ready. (עורך את השולחן= is setting the table) It's a difficult word to translate into English. דין is 'law'. I'm can't come up with a good way to translate both words together in a literal way, but hopefully that helps you understand the meaning.
That's true, but historically the combination was an accident - a mistake of Mishanic Hebrew that started with the Greek "archi". For the gory details, if you can read Hebrew, https://hebrew-academy.org.il/2010/11/08/%D7%A2%D7%95%D7%A8%D7%9A-%D7%93%D7%99%D7%9F/.
No. Both אבא שלי and האבא שלי are equivalent in modern Hebrew. Without ה is more common. It's place in the sentence requires it to be definite; the word אבא in itself hovers in modern Hebrew between being definite (as it was in its original Aramaic) and indefinite.
(The vav doesn't matter actually. If you'd say האבא, you'd say והאבא. The prepositions מ and ב swallow the ה, but ו does not.)