A barátom gyorsan megy a színházba, mert késik.

In Hungarian, possessed nouns (like barátom – my friend) almost always take a definite article (a / az) when they are normal subjects or objects.

• A barátom gyorsan megy… = My friend goes quickly…
• Leaving out the article (Barátom gyorsan megy…) is possible but sounds more poetic, emphatic, or unusual in everyday speech.

So in neutral, everyday Hungarian, A barátom is the standard way to say my friend as a subject here.

Barátom is made of:

• barát = friend
• -om = my (1st person singular possessive suffix)

So barátom literally = friend-my → my friend.

Hungarian usually uses possessive suffixes instead of separate words like my, your, etc.

Gyors is an adjective (fast, quick) that describes a noun:

• gyors autó = a fast car

Gyorsan is the adverb form (quickly, fast) that describes a verb:

• gyorsan megy = (he/she) goes quickly

In Hungarian, adverbs of manner are often formed by adding -an / -en to the adjective:

• lassú → lassan = slow → slowly
• gyors → gyorsan = fast → quickly

Megy is the 3rd person singular present tense of menni (to go):

• én megyek – I go
• te mész – you go
• ő megy – he/she goes

The older form megyen exists in historical or very old-fashioned style, but in modern standard Hungarian you just say megy.

Also, Hungarian doesn’t have a separate “is going” tense. Megy can mean:

• He/she goes (habitual)
• He/she is going (right now)

Context tells you which is meant.

The endings show different directions/locations:

• -ba / -be = into something (direction, to/into)
  
  • a színházba megy = (he/she) goes to the theatre (into the building)
• -ban / -ben = inside something (location, in)
  
  • a színházban van = (he/she) is in the theatre

In the sentence, your friend is moving towards/into the theatre, so színházba (illative case: into) is correct.

Hungarian almost always uses a definite article (a / az) with specific, known places:

• megyek a boltba = I’m going to the shop
• megy a színházba = (he/she) is going to the theatre

Leaving out the article (megy színházba) is possible in special, fixed expressions or styles, but the neutral, everyday form is megy a színházba.

The basic neutral order is:

• A barátom gyorsan megy a színházba…
  Subject – adverb – verb – place

You can say A barátom a színházba megy gyorsan, A barátom megy gyorsan a színházba, etc., but word order in Hungarian changes the focus/emphasis:

• A barátom gyorsan megy a színházba…
  Neutral: he’s going there quickly.
• A barátom a színházba megy gyorsan…
  Emphasis more on to the theatre (not somewhere else) as the place he is quickly going to.

So your original sentence is the most neutral and natural version.

It is your friend who is late.

• A barátom gyorsan megy a színházba, mert késik.
  = My friend is going quickly to the theatre because he/she is late.

The subject a barátom carries over to the second verb késik. Hungarian doesn’t repeat the pronoun ő here, because it’s clear the same person is being talked about.

Hungarian usually drops subject pronouns when they are clear from context or verb endings.

• (Ő) késik. = He/She is late.
• A barátom gyorsan megy a színházba, mert (ő) késik.

Adding ő is possible, but in this sentence it would add extra emphasis:

• … mert ő késik.
  = because HE/SHE is late (as opposed to someone else).

The neutral, un-emphasized form is without ő.

Késik is a verb, not an adjective.

• Infinitive: késni = to be late
• késik = (he/she) is late / arrives late

So:

• Ő késik. = He/She is late.
• A busz késik. = The bus is late.

There is also the adjective késő (late), but you don’t say ő késő to mean “he is late” in the sense of “arrives late”. You use the verb késik.

Yes, you can, but there is a nuance:

• mert késik
  = because he/she is late (generic: is running late, is in a state of lateness)
• mert elkésik
  = because he/she will end up being late / will be late in the end
  (el- adds a sense of completion/result)

In your sentence, mert késik is perfectly natural and means he/she is already late (or currently running late), and that’s why they’re hurrying.

Mert means because and introduces a subordinate clause explaining the reason.

Hungarian punctuation normally separates such a clause with a comma:

• A barátom gyorsan megy a színházba, mert késik.
• Elmegyek, mert fáradt vagyok. = I’m going away because I’m tired.

So the comma before mert is standard and correct.