Auf dem Tisch liegt das Wörterbuch, und darunter liegt der Merkzettel.

Because auf is a two-way preposition.

• Dative is used for location: where something is
  
  • auf dem Tisch = on the table
• Accusative is used for movement toward a place: where something is going
  
  • auf den Tisch = onto the table

Here nothing is moving onto the table; the dictionary is already there, so German uses dative: dem Tisch.

German main clauses follow the verb-second rule. That means the finite verb must be the second element in the clause.

So if you begin with Auf dem Tisch, that takes the first position, and the verb must come next:

• Auf dem Tisch | liegt | das Wörterbuch

A more basic order would be:

• Das Wörterbuch liegt auf dem Tisch.

Both are correct. Starting with Auf dem Tisch puts more emphasis on the location.

German often uses position verbs where English simply uses to be.

• liegen = to lie, to be lying
• stehen = to stand, to be standing
• sitzen = to sit, to be sitting

For flat objects like a dictionary or a note resting on a surface, liegen is very natural. So:

• Das Wörterbuch liegt auf dem Tisch
  = literally The dictionary lies on the table

In natural English, we usually just say The dictionary is on the table, but German prefers liegen here.

Darunter means under it, under that, or beneath it.

It is referring back to something already mentioned. In this sentence, it most naturally means under the dictionary.

So the second clause means:

• und darunter liegt der Merkzettel
  = and underneath it lies the note

German often uses words like darunter instead of repeating the noun.

Because it is a very common German structure called a pronominal adverb.

German often combines:

• da(r)-
  • preposition

Examples:

• darauf = on it
• darunter = under it
• damit = with it
• davon = of it / from it

The r appears when it helps pronunciation, especially before a vowel sound in the preposition. With unter, the standard form is simply darunter.

Because der Merkzettel is the subject, not an object.

The verb liegen does not take a direct object here. It is an intransitive verb. So the thing doing the “lying” is the subject:

• der Merkzettel liegt

Even though the word order changes, the case tells you the function:

• der Merkzettel = nominative masculine singular

So der Merkzettel is the subject of the second clause.

Because in German, all nouns are capitalized.

So:

• der Tisch
• das Wörterbuch
• der Merkzettel

This is a standard rule in German spelling, not something special about this sentence.

Because the sentence joins two main clauses:

• Auf dem Tisch liegt das Wörterbuch
• darunter liegt der Merkzettel

In German, with und, a comma between two main clauses is often optional. Many writers leave it out, and many include it for clarity or rhythm.

So both of these are acceptable:

• Auf dem Tisch liegt das Wörterbuch und darunter liegt der Merkzettel.
• Auf dem Tisch liegt das Wörterbuch, und darunter liegt der Merkzettel.

Because the second part is a complete main clause of its own, and a main clause needs a finite verb.

So German says:

• ..., und darunter liegt der Merkzettel.

Repeating liegt is normal and natural. German does not usually drop it here in standard full sentence structure.

You usually have to learn the gender with each noun, but compounds give you a useful clue: the last part determines the gender.

• das Wörterbuch: the last part is Buch, which is neuter, so the whole word is das Wörterbuch
• der Merkzettel: the last part is Zettel, which is masculine, so the whole word is der Merkzettel

This is a very helpful rule for German compound nouns.

Yes. That would also be correct.

• Unter dem Wörterbuch liegt der Merkzettel.
  = The note is under the dictionary.

Using darunter is just a more compact way to avoid repeating dem Wörterbuch. German does this very often.

So both are possible:

• ... und darunter liegt der Merkzettel
• ... und unter dem Wörterbuch liegt der Merkzettel

The version with darunter sounds smoother because it avoids repetition.