Hún kom grátandi heim eftir samtalið.

Kom is the 3rd-person singular past tense of koma (to come).

So:

• ég kem = I come
• hún kemur = she comes
• hún kom = she came

It matches hún because the subject is she.

Grátandi is the present participle of gráta (to cry / to weep).

In this sentence, it means something like:

• crying
• while crying
• in tears

So kom grátandi means she came in a crying state, or she came while crying.

It is participial in form, but here it works adverbially: it describes how she came home.

So the structure is basically:

• Hún kom = She came
• grátandi = crying / while crying
• heim = home

It does not mean that grátandi is a fully conjugated verb here. The main finite verb is only kom.

Because -andi participle forms are normally indeclinable in this kind of use. That means the form stays the same instead of changing for gender, number, or case.

So you can get forms like:

• Hún kom grátandi
• Hann kom grátandi
• Þau komu grátandi

The participle stays grátandi.

This is a very common Icelandic distinction:

• heim = homeward / to home (movement)
• heima = at home (location)

So:

• Hún kom heim = She came home
• Hún er heima = She is at home

In your sentence, there is movement toward home, so heim is the correct form.

Because heim already contains that directional idea. Icelandic often uses special adverbs for places where English would use to + noun.

So English says:

• come home

Icelandic says:

• koma heim

You do not need til here. In fact, til heim would be wrong in this meaning.

Samtalið is the noun samtal (conversation) with the definite article attached.

• samtal = a conversation
• samtalið = the conversation

In Icelandic, the word the is usually added to the end of the noun instead of being a separate word.

Here eftir means after, in a time sense: after the conversation.

With this meaning, eftir takes the accusative, and the accusative singular definite form of samtal is samtalið.

So:

• eftir samtal = after a conversation
• eftir samtalið = after the conversation

This is a useful thing to remember: some Icelandic prepositions require a specific case, and eftir can take different cases depending on meaning.

Not necessarily. It usually means that she was crying when she came home.

So the focus is on her state during the action of coming home, not on the exact moment the crying began.

If you wanted to say she started crying, Icelandic would normally use a different construction, such as a verb meaning begin.

Yes. Icelandic word order is somewhat flexible, though not completely free.

This sentence is very natural:

• Hún kom grátandi heim eftir samtalið.

You could also say:

• Hún kom heim grátandi eftir samtalið.
• Eftir samtalið kom hún grátandi heim.

The last version puts emphasis on after the conversation.

But remember the Icelandic verb-second pattern: if you move something to the front, the finite verb usually still comes second:

• Eftir samtalið kom hún ... not
• Eftir samtalið hún kom ...

Yes, very much so. Koma heim is the normal way to say come home.

It is one of those everyday combinations that learners see often, just like:

• fara heim = go home
• koma heim = come home
• vera heima = be at home

So in this sentence, kom heim is a very natural and idiomatic unit.