Learn English – Would it be wrong to write “I’ll pay the next time we’ll have dinner” instead of “”I’ll pay the next time we have dinner”

These distinctions are simply guidelines but not hard-and-fast rules. The shift of tense does not indicate that the event has not taken place.

Let's say that she agreed last week to have lunch with Susan today, to discuss a business proposition. The lunch does take place. She has a sandwich. Susan has soup. If the question is about what she said, it doesn't matter whether the lunch took place or not:

When I spoke with her on the phone, she said she would have lunch with Susan today, and they did indeed have lunch. She's a woman of her word.

Not needing to change the tense when the assertion is still true does not imply the corollary that changing the tense means the assertion is now false:

He said his brother worked in Milan.

His brother may still work in Milan. The shift of tense signals that the speaker is reporting the speech of another, not making the assertion himself.

This is an older meaning of "as" that is now only found in some dialects. It is a relative conjunction, or perhaps a relative pronoun, and it means "that". It is not standard English (so don't use it). Standard English uses "that".

It is sense 9 in the wiktionary definition as

Rowling uses this to establish the character of Hagrid as it a marker of region (Hagrid is from the West Country), and class (Dumbledore was also West Country or Gloucestershire, but had lost the accent). It marks Hagrid as very different from the way people speak in Little Whinging in Surrey.

Hagrid also uses the object form "them" as the complement of the verb "is". This is standard English, though using the subject form "they" would also be correct. (Saying "It is they" applies Latin grammar, which does always use the nominative case for this)