Breakdown of Я хочу прочитать пыльный дневник, который вы нашли.
я
I
вы
you (plural or formal)
хотеть
to want
прочитать
to read (perfective)
Questions & Answers about Я хочу прочитать пыльный дневник, который вы нашли.
Since it is the direct object of the second clause ("you found which"), shouldn't который have an accusative case ending?
It actually is in the accusative case! Который refers back to дневник (diary), which is a masculine noun. Because a diary is inanimate (not alive), the masculine accusative form for both the noun and который looks exactly the same as their basic nominative forms.
Can I drop который like we drop "which" in English and just say ...пыльный дневник, вы нашли?
No, you can never skip the relative pronoun in Russian. You must always include который (or its matching form) to connect the two parts of the sentence, and it must always be preceded by a comma.
What would happen to который if we were talking about a feminine noun, like книга (book)?
Который must match the gender of the noun it replaces. For a feminine noun in the accusative case, the ending changes to -ую. The sentence would become: Я хочу прочитать пыльную книгу, которую вы нашли.
Why use прочитать here instead of читать?
Прочитать is a perfective verb, which focuses on successfully completing an action. The speaker wants to read the diary all the way through to the end. If they just wanted to "spend some time reading" without necessarily finishing it, they would use the imperfective verb читать.