Learn Hardcore German: Die Lampe an der Wand ist die hellste im ganzen Haus. - The lamp on the wall is the brightest in the whole house.

Lampe is a feminine noun in German. In the nominative singular (subject position), feminine nouns take the definite article die. So die Lampe simply means “the lamp.”

Why is an der Wand in the dative case?

The preposition an is a Wechselpräposition (two-way preposition). It takes:

accusative when indicating motion toward something (Wo‑hin?),
dative when indicating location/static position (Wo?).
Here we describe where the lamp is (“on the wall”), so we use the dative feminine article der: an der Wand.

Why does the sentence use die hellste instead of just hellste?

In die hellste, hellste is a nominalized adjective in its superlative form (“brightest one”). Whenever you turn an adjective into a noun in German, you need the appropriate article plus the adjective’s ending:

feminine nominative article: die
superlative ending for nominalized adjectives: -e
So die hellste = “the brightest (one).”

Could we say ist am hellsten im ganzen Haus instead? What’s the difference?

Yes. am hellsten is the adverbial superlative (“most brightly”). The two versions differ in nuance:

die hellste = nominal: “the brightest one” (focus on the lamp as an item)
am hellsten = adverbial: “shines most brightly” (focus on the action/extent of brightness)

Why is there im instead of in dem, and why does ganz become ganzen?

im is simply the contraction of in dem (used before masculine or neuter dative).
After dem (dative definite article), adjectives take the weak ending -en. Since Haus is neuter and in dative, in dem ganzen Haus contracts to im ganzen Haus.

Why do we say das ganze Haus in some sentences but im ganzen Haus here?

das ganze Haus is nominative neuter (“the whole house”) when ganz is attributive and not inside a prepositional phrase.
Inside in dem Haus (dative), ganz must take the dative weak ending -en, giving im ganzen Haus.