Learn Hardcore Hungarian: A barátom futva megy a buszmegállóhoz, mert késik. - My friend goes to the bus stop running because he is late.

In Hungarian, possessed nouns normally take the definite article a/az in neutral sentences: a barátom = “my friend.”
The ending -om marks “my,” and the article a just marks that this is a specific, known person.
You usually drop the article with possessed nouns only in special cases, e.g. direct address (Barátom! “My friend!”) or some fixed expressions.
So A barátom futva megy… is the natural, neutral way to say “My friend is going…”

What does the ending -om in barátom mean exactly?

The ending -om is the 1st person singular possessive ending (“my”).
So:

barát = friend
barátom = my friend
Other examples: ház → házam (“my house”), könyv → könyvem (“my book”).
Because the verb forms show who the subject is, you do not need to add én; A barátom megy already clearly means “My friend is going.”

What is futva? Why is it futva megy and not just fut?

Futva is the -va/-ve adverbial participle of fut (“to run”). It literally means “runningly / while running,” and it describes how the action is done.
The structure [verb]-va/-ve + megy is very common:

gyalog megy – “goes on foot”
sietve megy – “goes, hurrying”
futva megy – “goes, running”

So A barátom futva megy is like “My friend goes, running,” i.e. “My friend goes to the bus stop running / runs to the bus stop.”

Could I say A barátom fut a buszmegállóhoz instead? What is the difference from futva megy a buszmegállóhoz?

Yes, A barátom fut a buszmegállóhoz is correct and means almost the same: “My friend runs to the bus stop.”
Nuance:

fut a buszmegállóhoz focuses more directly on the action “runs (to the stop).”
futva megy a buszmegállóhoz feels a bit more like “he goes there, and the manner is that he is running,” a bit more descriptive/neutral in style.

In everyday speech both are fine; fut a buszmegállóhoz is shorter and very natural, futva megy sounds slightly more “structured” or descriptive.

What does the ending -hoz in buszmegállóhoz mean?

The ending -hoz/-hez/-höz is the allative case, meaning “to / towards (a place).”
It attaches to the whole noun phrase, including the last element of a compound:

buszmegálló = bus stop
buszmegállóhoz = to the bus stop

Choice of -hoz / -hez / -höz depends on vowel harmony:

-hoz after back vowels (a, o, u) — e.g. házhoz
-hez after front unrounded vowels (e, é, i, í) — e.g. székhez
-höz after front rounded vowels (ö, ő, ü, ű) — e.g. körhöz

When do I use -hoz/-hez/-höz instead of -ba/-be or -ra/-re?

Rough guide:

-hoz/-hez/-höz = to / towards a place or person (to someone’s place, to a stop, to a building as a point)
- orvoshoz megyek – I’m going to the doctor
- iskolához megyek – I’m going to the school (to its vicinity)
-ba/-be = into something (interior)
- iskolába megyek – I’m going into school (as a student)
-ra/-re = onto / to the surface of / to an event place
- szigetre megyek – I’m going to the island
- koncertre megyek (usually koncertre / koncertre → in practice koncertre or koncertre with linking, but you’ll see koncertre megyek) – I’m going to the concert

For buszmegálló, Hungarian treats it as a point/place you go to, so buszmegállóhoz megyek is the natural choice.

Why is buszmegálló written as one word?

Hungarian usually writes compounds as one word. Buszmegálló is a compound of:

busz – bus
megálló – (a place where something stops)

Together: buszmegálló = bus stop.
Because the whole thing is one lexical unit, case endings go on the end: buszmegállóhoz (“to the bus stop”), buszmegállónál (“at the bus stop”), etc.

Why is there a comma before mert in …, mert késik?

Mert introduces a subordinate clause of reason (“because”), and in Hungarian you normally separate such clauses with a comma.
So:

A barátom futva megy a buszmegállóhoz, mert késik.

In general, conjunctions like mert, hogy, ha, amikor introducing a full clause are preceded by a comma when they link two clauses.

What exactly does késik mean here? Is it “is late” or “will be late”?

Késik is the 3rd person singular of késik (“to be late”). Literally: “(he) is late.”
Hungarian often uses the present tense where English uses a future-like meaning, based on context.
So mert késik can mean:

“because he is (running) late now”
or, in context:
“because he will be late (otherwise)” / “because he’s going to be late.”

The language does not force you to mark the future here; context gives the time reference.

Why is there no word for “he” in the sentence? Where is the subject “he”?

Hungarian usually drops personal pronouns when they are not needed for emphasis, because the verb ending already shows the person and number.

késik = “he/she/it is late” (3rd person singular)
The subject A barátom (“my friend”) is explicitly mentioned, so adding ő (“he/she”) would be unnecessary and untypical here.
You’d use ő only for contrast or emphasis: Ő késik, nem én. – “HE is late, not me.”

Could I say mert késni fog instead of mert késik? What is the difference?

Yes, mert késni fog is grammatical and more explicitly future: “because he will be late.”
Difference:

mert késik – very natural, neutral; present tense, but can describe a near-future result (“he’s running late / he’s going to be late”).
mert késni fog – emphasizes the future aspect: the lateness will occur later. It can sound a bit more formal or more “planned/expected,” depending on context.

In everyday speech, mert késik is usually preferred unless you really need to stress the future.

Is késik related to elkésik? Which one should I use for “to be late”?

Yes, elkésik is the prefixed form of késik.

késik – “is late / is running late” (state)
elkésik – “ends up being late, (finally) is late (for something)” (more event-like)

In simple sentences about being late in general, both are common:

Mindig késik. – He is always late.
Mindig elkésik. – He always ends up being late.

In your sentence, mert késik is perfectly natural; mert el fog késni would emphasize the event of ending up late.

Can I change the word order, for example say A barátom a buszmegállóhoz futva megy, mert késik or A barátom a buszmegállóhoz megy futva?

Hungarian word order is flexible, but changes in order can change focus or sound unnatural.

A barátom futva megy a buszmegállóhoz, mert késik. – neutral, topic (A barátom), then predicate.
A barátom a buszmegállóhoz futva megy, mert késik. – still understandable, but a bit clumsy; too much material between megy and its complement.
A barátom a buszmegállóhoz megy futva – possible in speech with special emphasis on the manner (futva), but neutral Hungarian prefers keeping futva before megy.

For learners, it’s safest to keep [subject] + [manner] + [verb] + [destination]:
A barátom futva megy a buszmegállóhoz.

Could I use mivel instead of mert here? What is the difference?

You could say …, mivel késik, and it would still mean “because he is late.”
Nuance:

mert – the default, neutral “because,” used in almost all contexts.
mivel – also “because,” but often sounds slightly more formal or explanatory, as if you’re giving a justification or background.

In everyday speech, mert is much more common here:
A barátom futva megy a buszmegállóhoz, mert késik.