Learn Hardcore Danish: Mit yndlingsfag er musik, fordi jeg elsker rytmen. - My favorite subject is music because I love the rhythm.

Danish has two grammatical genders for nouns: common gender and neuter gender.
The possessive pronouns agree with the gender and number of the noun they describe:

min – for singular common-gender nouns (en-words), e.g. min bog (my book)
mit – for singular neuter-gender nouns (et-words), e.g. mit fag (my subject)
mine – for all plural nouns, e.g. mine fag (my subjects)

The word fag is neuter (you say et fag), so you must use mit. Therefore mit yndlingsfag is correct, not min yndlingsfag.

Why is yndlingsfag written as one long word instead of two words?

Danish, like other Scandinavian languages and German, loves compound nouns.
You often join two nouns into one word:

yndling (favourite) + fag (subject) → yndlingsfag (favourite subject)

Writing it as two separate words (yndlings fag) would be wrong and look very strange to a Dane.
So whenever something is “X-subject”, “X-book”, etc., Danish usually sticks the words together: fodboldkamp, skrivebord, sommerferie, and here yndlingsfag.

What exactly does yndlingsfag mean? Is yndling a word by itself?

Yes, yndling is a word by itself. Roughly:

yndling – favourite (as a noun), like pet or favourite person/thing
fag – subject (in school), or field of study

So yndlingsfag literally means favourite subject (school subject).
You can also find yndlings- as a prefix in other compounds: yndlingsfarve (favourite colour), yndlingssang (favourite song), yndlingsfilm (favourite movie).

Why is there no article before musik? Why not en musik or et musik?

Musik is usually an uncountable noun in Danish when you talk about music in general.
Uncountable nouns in a general sense normally take no article:

Jeg kan godt lide musik. – I like music.
Han drikker kaffe. – He drinks coffee.
Vi laver mad. – We cook (food).

You could use an article with musik in some special contexts (for example talking about a specific piece or type of music), but in this sentence you mean music in general, so just musik is correct.

Could you also say Musik er mit yndlingsfag instead of Mit yndlingsfag er musik?

Yes, both sentences are grammatically correct, but the emphasis changes a bit.

Mit yndlingsfag er musik focuses on what your favourite subject is.
Musik er mit yndlingsfag focuses on music and tells us that, out of the subjects, music is the one you like best.

Both follow normal Danish word order rules (finite verb in second position in a main clause):
Mit yndlingsfag (position 1) er (verb, position 2) musik (rest).
Musik (position 1) er (verb, position 2) mit yndlingsfag (rest).

Why is there a comma before fordi? Is that always required?

Modern Danish allows two systems:

Grammatical comma (traditional): you must put a comma before most subordinate clauses, including those starting with fordi.
Pause comma (optional system): the comma is based more on spoken pauses; you can sometimes leave it out.

Most teaching materials and schools still prefer the grammatical comma, so they write:
Mit yndlingsfag er musik, fordi jeg elsker rytmen.

So: in standard written Danish for learners, it is correct (and safest) to keep the comma before fordi.

Why is the word order fordi jeg elsker and not something like fordi elsker jeg?

Fordi introduces a subordinate clause (a dependent clause).
In Danish subordinate clauses, the normal order is subject–verb:

fordi jeg elsker rytmen – because I love the rhythm

In main clauses, Danish has verb-second word order, so the finite verb usually comes in second position. But after a subordinating conjunction like fordi, you generally do not use verb-second; you use subject–verb.

So fordi elsker jeg rytmen is incorrect in standard Danish.

What is the difference between elsker and kan lide in this kind of sentence?

Both can translate to like, but elsker is much stronger:

jeg kan lide rytmen – I like the rhythm
jeg elsker rytmen – I love the rhythm (much more enthusiastic)

Using elsker about people can mean romantic love or strong affection.
Using elsker about things or activities (music, rhythm, football, chocolate) is very common and expresses strong liking or passion.
In this sentence, elsker emphasizes that you are really passionate about the rhythm in music.

Why is it rytmen and not just rytme?

Rytmen is the definite form of rytme.

en rytme – a rhythm
rytmen – the rhythm

In the sentence, you are talking about the rhythm that exists in music in general, so Danes normally use the definite form here: jeg elsker rytmen.
Danish usually expresses the by attaching -en / -et to the end of the noun, not with a separate word.

Would jeg elsker rytme (without -n) also be possible?

It is possible, but it sounds less natural here and changes the nuance.

jeg elsker rytme sounds more like I love rhythm as a general concept (a bit abstract or stylistic).
jeg elsker rytmen feels like you are talking about the rhythm that is actually present in music you are hearing or playing.

Everyday Danish in this context would almost always use jeg elsker rytmen.

Could you leave out fordi and just say Mit yndlingsfag er musik, jeg elsker rytmen?

In spoken, informal Danish, you might hear people chain clauses like that, but in standard written Danish it is considered wrong or at least poor style.
You should link the clauses with something, usually:

Mit yndlingsfag er musik, fordi jeg elsker rytmen.

Without fordi, you get two main clauses just stuck together with a comma (a comma splice), which is normally avoided in correct writing.

Are there any tricky pronunciation points in this sentence for an English speaker?

Yes, a few:

mit – short, clipped i, a bit like the i in hit, not like meet.
yndlingsfag – the y is like the French u in tu; lips rounded but tongue like i. The g in fag is often pronounced softly, almost like fa with a slight g at the end.
fordi – the r is the Danish uvular r (in the throat), and the di is like dee.
jeg – often pronounced more like yai or even yai/yi in fast speech.
elsk‑ in elsker – clear e (like e in bed), and the r at the end is weak.
rytmen – again the rounded y sound; ryt a bit like rüt, and -men is like mun.

Listening to natives and repeating this full sentence several times will help you get used to the vowel y and the soft final r.