I think "throughout" stresses that it takes the whole summer, right? במשך does not commit to it so strongly, I believe like "during". But it's a nuance, במשך (or "during") does suggest at least a significant part of the summer.
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?