I think it needs one more "the" (I'm using the Android app where you select from pre-typed boxes) - it reads better as "I am reading the book during THE summer"
How do you differentiate whether הקיץ means this summer or the summer? I think in this case it is this because it makes most sense in the context of the sentence.
Context is the key.
הקיץ .ה = the .קיץ = summer
For 'beintayyim', could "throughout" be just as good a translation as 'during'
I would say that "במדך" is a preposition here, not an adverb.
Yes, formally בְּמֶ֫שֶׁךְ is a preposition with a noun in the construct state ("in the duration of"), which is used de facto as a prepositional expression. Who thinks of it as an adverb, since it cannot stand alone?
May I use "throughout" for "during"?
I´d prefere to say:
"אני קורא ספר במשך הקיץ."
"אני קורא את הספר הזה במשך בקיץ.