What are AWAY and APART modifying here?

Solution 1:

Some call away and apart adverbs. They can be adverbs, but they don't always work that way, and they have different meanings and grammar.

All the examples of away and apart in the Original Question occur in measure phrases -- miles apart, two weeks away, three days apart, etc. These phrases involve a quantifier, either overt like two or understood like (several) days, and a quantity like distance, time, weight, and optionally (as here) something that indicates specifics about the quantity that is being measured.

This something extra can be away, which means away from some zero point like now or here, and refers to the distance to be covered or the time to be expended (or the fare money to be collected, as in Chicago is $18 away) in traveling to a place. Away is asymmetric -- it only goes in one direction:

• Scranton is 30 minutes away (from Utopia)
  means it takes 30 minutes to go (from Utopia) to Scranton. However,
• *Scranton and Utopia are 30 minutes away
  is ungrammatical. Away requires a singular referent, not plural.

This extra thing can also be apart (as well as long, wide, tall, old, and many more). Apart comes from part and it is used to refer to things that occur together, as in the idioms come apart, rip apart, fall apart, etc.

In a measure phrase, apart requires a plural referent:

• Scranton and Utopia are 30 minutes apart
  is perfectly grammatical, but
• *Scranton is 30 minutes apart from Utopia
  is not.

As for what modifies what, don't worry about it.
The definition of adverbs they gave you is full of holes.

Solution 2:

Actually, they aren't adverbs—despite what you might read in a dictionary.

There are a number of clues here:

• they primarily describe spatial relations, but can also be used to describe temporal ones
• they can be used to describe noun phrases, and can exist as Complements of the verb BE
• they can be modified by the specialised adverbs right and straight
• they don't have any typical adverbial suffix and don't have any homophonous adjectives either
• the phrases they head can postmodify nouns and noun phrases
• they can't be straightforwardly modified by the adverb very

One atypical property they have is that they don't occur before noun phrases. This is because they are intransitive prepositions.

In the Original Poster's examples they do not modify anything—and, indeed, this is an important point. It's because they are Locative Complements of the verb BE, and not Modifiers, that we know they are not adverbs.

For more information about intransitive pepositions, see either of the following:

• The Cambridge Grammar of the English Language, Rodney Huddleston & Geoffrey Pullum, 2002.

• Oxford Modern English Grammar, Bas Aarts, 2014.