Learn Hardcore German: Diese Lampe ist so hell wie der Mond.

Lampe is grammatically feminine in German: die Lampe.
The demonstrative dies- (“this”) has to match the noun in gender, case, and number.

Feminine, nominative, singular: diese Lampe
Masculine, nominative, singular would be: dieser Tisch
Neuter, nominative, singular: dieses Buch

In the sentence, „Diese Lampe“ is the subject (nominative), so the correct form is diese, not dieser.

Why is the verb „ist“ in second position?

German main clauses follow the verb-second (V2) rule: the conjugated verb must be in second position, no matter how long the first element is.

First position: Diese Lampe
Second position: ist
Rest of the sentence: so hell wie der Mond

You could put something else first, but the verb still stays second, for example:
So hell wie der Mond ist diese Lampe.

Why is it „hell“ and not „helle“ or „helles“?

In „ist so hell“, hell is a predicative adjective (it comes after the verb sein and describes the subject).
Predicative adjectives in German do not take endings:

Die Lampe ist hell.
Der Mond ist schön.
Die Sterne sind sichtbar.

Adjective endings (helle, heller, helles, etc.) appear when the adjective comes before a noun:

die helle Lampe
ein heller Mond

How does the structure „so … wie“ work here?

„so … wie“ is the standard way to express equality in German:

so + adjective/adverb + wie + comparison term

In the sentence:

so hell = as bright
wie der Mond = as the moon

Other examples:

Er ist so groß wie sein Bruder. – He is as tall as his brother.
Sie spricht so schnell wie du. – She speaks as fast as you.

What is the difference between „so … wie“ and „… als“?

Use:

„so … wie“ for equality:
- Die Lampe ist so hell wie der Mond. – The lamp is as bright as the moon.
Adjective in -er + als for inequality (comparative):
- Die Lampe ist heller als der Mond. – The lamp is brighter than the moon.

So:

so hell wie → same degree
heller als → a higher degree

Why is it „wie der Mond“ and not „wie den Mond“?

The case after „wie“ depends on the implicit clause you can imagine behind it.
You can think of it as:

Diese Lampe ist so hell, wie der Mond (hell ist).

In that imaginary clause, der Mond is the subject of ist, so it must be nominative: der Mond, not den Mond.

If the compared element would be an object in the implied clause, you’d use another case, e.g.:

Er liebt sie so sehr wie mich.
→ (… wie er mich liebt.) – mich is accusative.

Why is „Mond“ capitalized, but „hell“ is not?

In German:

All nouns are capitalized: der Mond, die Lampe, das Haus.
Adjectives are normally not capitalized: hell, groß, schön.

So Mond is a noun → capital letter.
hell is an adjective → lowercase.

Why do we say „der Mond“ and not just „Mond“ with no article?

German generally places a definite article in front of many nouns where English might omit it, especially with:

Celestial bodies: die Sonne, der Mond, die Sterne
Body parts: die Hand, der Kopf, etc. (in many contexts)

So in most normal sentences, Germans say:

Der Mond ist hell. – “(The) moon is bright.”

Leaving the article off („Mond ist hell“) sounds ungrammatical in standard German.

Can I change the word order, for example: „Diese Lampe ist wie der Mond so hell“?

No, that sounds unnatural or wrong in standard German.
In the „so … wie“ structure, so and its adjective belong tightly together, and wie directly introduces the comparison term:

Natural: Diese Lampe ist so hell wie der Mond.
Also natural: Diese Lampe ist genauso hell wie der Mond.

Splitting it into „… wie der Mond so hell“ breaks the normal pattern and is not idiomatic.

Can I say „Diese Lampe ist hell wie der Mond“ without „so“?

You can hear „hell wie der Mond“, but it tends to sound more like a simile (“bright like the moon”) rather than a precise equality comparison.

so hell wie der Mond → clearly “as bright as the moon” (same degree)
hell wie der Mond → stylistic/figurative “bright like the moon”

For learners, it’s safer and more standard to use „so … wie“ when you mean strict “as … as”.