Learn Hardcore German: Die Bibliothekarin öffnet das Buch vorsichtig auf dem Tisch. - The librarian opens the book carefully on the table.

In German all nouns are capitalized. The suffix -in on Bibliothekarin marks a female person (feminine gender), and the nominative singular feminine definite article is die.

How can I tell that das Buch is the direct object, even though das also marks the nominative neuter form?

German neuter definite articles are das in both nominative and accusative. You recognize das Buch as the direct object because:

It follows the verb in the typical S-V-O order.
die Bibliothekarin (the subject) is performing the action öffnet on das Buch.
Context makes clear it’s what’s being “opened.”

Why is vorsichtig not inflected like an adjective before a noun, and what role does it play here?

Here vorsichtig is an adverb modifying the verb öffnet (it tells how she opens). German adverbs:

Share the form of the adjective’s base.
Take no endings.
If you used vorsichtig to describe a noun (e.g. ein vorsichtiges Vorgehen), you would inflect it as an adjective.

Why does auf dem Tisch use the dative case, and how do I know when auf takes dative or accusative?

auf is a two-way (Wechsel-)preposition:

Dative answers Wo? (location) → auf dem Tisch “on the table.”
Accusative answers Wohin? (direction) → auf den Tisch “onto the table.”
Since the book is opened on the table (static location), you use dative (dem).

Could I use separable aufmachen instead of öffnen for a book? What’s the difference between öffnen, aufmachen, and aufschlagen?

öffnen is the standard verb for “to open” objects, including books.
aufmachen (separable) is more colloquial and typically used for containers (doors, windows, bottles).
aufschlagen often means “to open (a book/newspaper) to a page” or “to slam open” something.
So for simply opening a closed book, öffnen is most natural; das Buch aufschlagen suggests flipping it open to read.

Why is the verb öffnet in second position and not at the end of the sentence?

In main clauses German follows the Verb-second (V2) rule:

One element (subject, adverbial, etc.) comes first.
The conjugated verb occupies the second slot.
Everything else follows.
Because this is a main clause, öffnet stays in position two.

Is the order of das Buch vorsichtig auf dem Tisch fixed, or can I rearrange these elements?

German word order is flexible, but the V2 rule still governs the verb’s position. You can front other elements:

Vorsichtig öffnet die Bibliothekarin das Buch auf dem Tisch.
Auf dem Tisch öffnet die Bibliothekarin vorsichtig das Buch.
However, the neutral sequence for manner and place is Manner → Place, so vorsichtig before auf dem Tisch feels most natural.

Where would a time expression fit if I wanted to say “This morning, the librarian opened the book carefully on the table”?

German commonly uses the Time-Manner-Place (TeMoLo) pattern. Insert the time phrase first:
Heute Morgen öffnet die Bibliothekarin das Buch vorsichtig auf dem Tisch.
Here Heute Morgen (time) → vorsichtig (manner) → auf dem Tisch (place).

How would I form a passive version of this sentence?

Use werden + past participle and optionally a von-phrase for the agent:
Das Buch wird von der Bibliothekarin vorsichtig auf dem Tisch geöffnet.

Das Buch becomes the grammatical subject.
wird geöffnet is the passive verb.
von der Bibliothekarin names the agent in dative.