Learn Hardcore Icelandic: Brúðkaupið er lítið en mjög fallegt.

Icelandic usually attaches the definite article as an ending instead of using a separate word like English the.
Brúðkaup is a neuter noun meaning wedding, and the neuter definite ending in the nominative singular is -ið.
So brúðkaup → brúðkaupið = the wedding.

What gender is brúðkaup, and how does that affect the sentence?

Brúðkaup is a neuter noun.
Because the subject is neuter singular, the describing adjectives (lítið, fallegt) also have to be in the neuter singular nominative form.
That is why you see -t at the end of both adjectives.

Why is it lítið and not lítill or lítil?

The base form of the adjective is lítill (m.), lítil (f.), lítið (n.) = small.
Since brúðkaupið is neuter, you must choose the neuter form: lítið.
If the subject were masculine or feminine, you would use lítill or lítil instead.

Why is it fallegt and not fallegur?

The base forms of this adjective are fallegur (m.), falleg (f.), fallegt (n.) = beautiful / pretty.
Again, because brúðkaupið is neuter, the correct agreeing form is fallegt.
So the pattern is:

fallegur maður (m.) – a handsome man
falleg kona (f.) – a beautiful woman
fallegt brúðkaup (n.) – a beautiful wedding

Why don’t the adjectives also show “the”? Why not something like hitt litla brúðkaup here?

In this sentence, lítið and fallegt are predicative adjectives (they come after the verb er, describing the subject).
In Icelandic, predicative adjectives do not take a definite ending; only the noun carries the article: brúðkaupið.
You see adjective definiteness mainly when the adjective comes before the noun, e.g. hið litla brúðkaup = the small wedding (literally the-the small wedding: article + definite adjective + indefinite noun).

What case is Brúðkaupið in, and why?

Brúðkaupið is in the nominative singular.
The nominative is used for the subject of the sentence, and here brúðkaupið is the thing that is small and beautiful.
The adjectives lítið and fallegt also appear in nominative to agree with the subject.

What exactly does en mean here? Could I use og instead?

Here en means but, showing a contrast: small but very beautiful.
Og means and, which simply adds things without contrast.
If you used og (… er lítið og mjög fallegt), it would sound more like small and very beautiful with less of a “surprisingly / although” feeling.

What does mjög do, and how is it different from mikið?

Mjög is an adverb meaning very, and it modifies adjectives and other adverbs:

mjög fallegt = very beautiful
mjög lítið = very small

Mikið is usually related to the idea of much / a lot (often as an adjective or adverb in other structures), not the standard intensifier for adjectives.
So mjög fallegt is the normal way to say very beautiful, not mikið fallegt.

Could I change the order of the adjectives, like Brúðkaupið er mjög fallegt en lítið?

Yes, you can say Brúðkaupið er mjög fallegt en lítið.
The meaning is almost the same, but the emphasis shifts slightly:

… er lítið en mjög fallegt highlights small first, then contrasts it with very beautiful.
… er mjög fallegt en lítið first stresses that it’s very beautiful, then adds the surprising fact that it’s small.

How do you pronounce Brúðkaupið er lítið en mjög fallegt?

Approximate IPA pronunciation is:

Brúðkaupið: [ˈpruːðˌkʰœiːpɪð]
er: [ɛr]
lítið: [ˈliːtɪð]
en: [ɛn]
mjög: [mjœːɣ] (the final sound is a soft, voiced [ɣ], like a relaxed g in the back of the mouth)
fallegt: [ˈfahtlɛxt] or [ˈfalːɛxt] depending on speaker/region

Key points:

ð is a soft th-sound [ð], like English this, but often quite weak.
au in kaup is [œi], a bit like English “uh-ee” blended.
gg / ll + consonant often produce special clusters; in fallegt you don’t pronounce a clear hard g.