Laura lähti kotoa ilman että hän ehti syödä aamiaista, koska taksi tuli liian aikaisin.

Ilman että means without when it is followed by a full clause.

• ilman
  • noun phrase: ilman aamiaista = without breakfast
• ilman että
  • clause: ilman että hän ehti syödä aamiaista = without her having time to eat breakfast

So että is needed because what follows is not just a noun, but a whole clause with its own subject and verb.

Lähteä means to leave / set off / depart, while mennä means to go.

In this sentence, the important idea is that Laura departed from home, so lähti is the most natural choice.

• Laura lähti kotoa = Laura left home
• Laura meni... would usually make the listener expect a destination or purpose, such as Laura meni töihin = Laura went to work

So lähti fits better because the sentence focuses on the point of departure.

Kotoa means from home.

It is a very common special location form of koti. The most useful set to learn is:

• kotona = at home
• kotoa = from home
• kotiin = home / to home

So:

• Laura lähti kotoa = Laura left home

Even though koti is a normal noun, these forms are so common that learners usually memorize them as a set.

Ehti is the past tense of ehtiä, which means to have time to, to manage to, or to get around to before something else happens.

So hän ehti syödä aamiaista means:

• she had time to eat breakfast
• or more literally, she managed to eat breakfast in time

In this sentence, the idea is that she left before there was enough time for breakfast.

Because ehtiä is followed by the basic infinitive, the first infinitive.

So the pattern is:

• ehti syödä = had time to eat
• ehti nähdä = had time to see
• ehti tehdä = had time to do

This is just the normal verb pattern of ehtiä. Finnish verbs often require a specific infinitive form after them, and ehtiä takes the basic one.

Here aamiaista is in the partitive. That is natural for two reasons.

First, with verbs like syödä, the partitive is often used when the action is seen as ongoing, incomplete, or not resulting in a completed whole.

Second, in this sentence she did not actually get to eat the breakfast, so the action is clearly not completed.

Compare:

• syödä aamiaista = eat breakfast, be eating breakfast
• syödä aamiainen / aamiaisen = eat the whole breakfast, finish the breakfast

So aamiaista fits the meaning better here.

Yes, Finnish often leaves subject pronouns out when the verb already shows the person. But here hän is included to make the subordinate clause explicit and clear.

So:

• ilman että hän ehti syödä aamiaista is fully clear and natural
• leaving out hän might be possible in some contexts, especially in speech, but it would sound less explicit

For learners, keeping hän here is a good model.

Because the sentence needs an adverb, not an adjective.

• aikainen = early, as an adjective
  
  • aikainen taksi = an early taxi
• aikaisin = early, as an adverb
  
  • taksi tuli aikaisin = the taxi came early

Here aikaisin describes the verb tuli (came), so the adverb is required.

And liian aikaisin means too early.

Yes. A more compact version would be:

Laura lähti kotoa syömättä aamiaista, koska taksi tuli liian aikaisin.

Here syömättä means without eating.

The original sentence is still completely correct and clear. It is just a little more explicit because it includes the idea of not having time with ehti. The shorter version focuses more simply on the fact that she left without eating breakfast.