Learn Hardcore German: Die Bibliothek im Zentrum ist größer als die kleine Schule nebenan. - The library in the center is bigger than the small school next door.

When in expresses a static location (“where?”), it takes the dative.
das Zentrum in the dative becomes dem Zentrum, and in everyday German in dem is contracted to im.

How do you form the comparative of groß, and why is it größer als rather than something like mehr groß?

German comparatives are formed by adding the suffix -er to the adjective stem, plus als before the thing you’re comparing.
So groß → größer and you say größer als (“bigger than”) rather than using mehr.

When should I use als versus wie in comparisons?

Use als for unequal comparisons (“bigger than”, “faster than”).
Use wie when you compare things that are equal (“as big as”, “as fast as”).

Why does kleine take an -e ending in die kleine Schule? Shouldn’t it be kleiner?

Because after a definite article in the nominative feminine singular, adjectives take the weak ending -e.
We have die (nom. fem.) + kleine Schule.

Why is the verb ist placed in second position in this sentence?

German main clauses follow the verb‑second (V2) rule: the finite verb comes second, no matter how long the first constituent is.
Here Die Bibliothek im Zentrum is first, ist is second.

What is nebenan, and how is it different from neben der Schule?

nebenan is an adverb meaning “next door” or “nearby.”
neben by itself is a preposition that requires a noun in a case (e.g. neben der Schule, dative).
Using nebenan lets you end the phrase succinctly with a location adverb.

Why are both Bibliothek and Schule preceded by die?

Both Bibliothek and Schule are feminine nouns in German.
In the nominative singular, the definite article for feminine nouns is die.