Min søster loggede ud for tidligt og gik glip af den sidste besked.

Because søster is an en-word (common gender) in Danish.

With possessives, Danish usually matches the noun’s gender/number:

• min for singular en-words
• mit for singular et-words
• mine for plurals

So:

• min søster = my sister
• mit hus = my house
• mine søstre = my sisters

In Danish, a possessive normally replaces the article, just like in English.

So you say:

• min søster = my sister

Not:

• den min søster

This works the same way with other possessives too:

• min bror = my brother
• hendes telefon = her phone
• vores bil = our car

Loggede ud is the past tense of at logge ud, which means to log out.

This works a lot like an English phrasal verb:

• at logge ud = to log out
• logger ud = logs out / am logging out
• loggede ud = logged out

The verb part changes form, but the particle ud stays separate in normal sentence use.

So:

• Hun logger ud = She logs out
• Hun loggede ud = She logged out

Because they come from two different types of verbs.

• logge is a regular verb, so its past tense is formed with -ede:
  logge → loggede
• gå is irregular, so its past tense is:
  gå → gik

This is similar to English:

• log → logged = regular
• go → went = irregular

So in the sentence:

• loggede ud = logged out
• gik glip af = missed

Because here the word is being used adverbially: it describes when she logged out, not a noun.

• tidlig = early, used as an adjective
• tidligt = early, used as an adverb

Examples:

• en tidlig morgen = an early morning
• Hun kom tidligt = She arrived early

So:

• for tidligt = too early

The word for here means too, not for in the sense of purpose.

At gå glip af is a fixed expression meaning to miss or to miss out on something.

In the past tense, it becomes:

• gik glip af = missed / missed out on

So:

• Hun gik glip af bussen = She missed the bus
• Jeg gik glip af mødet = I missed the meeting

It is best learned as one whole expression:

• at gå glip af noget = to miss something

The literal meaning does not help much, so treat it as an idiom.

Because af is part of the fixed idiom at gå glip af.

You cannot normally leave it out. The full pattern is:

• gå glip af + noun

Examples:

• gå glip af filmen = miss the film
• gå glip af chancen = miss the opportunity
• gå glip af den sidste besked = miss the last message

So af belongs to the expression, not separately to besked.

Because Danish normally uses double definiteness when a definite noun has an adjective before it.

That means:

• den + adjective + noun

So:

• den sidste besked = the last message

Compare:

• beskeden = the message
• den sidste besked = the last message

In standard Danish, you usually do not say:

• sidste beskeden

And sidste besked without den would usually sound indefinite or more like a label/title, not the last message in normal sentence use.

Because besked is an en-word, not an et-word.

Definite forms with adjectives must match the noun’s gender:

• den for en-words
• det for et-words

So:

• den sidste besked = the last message
• det sidste brev = the last letter

Since en besked is correct, the sentence needs den sidste besked.

It has one subject and two coordinated past-tense verbs:

• Min søster = subject
• loggede ud = first verb phrase
• for tidligt = adverbial
• og = and
• gik glip af den sidste besked = second verb phrase

So the structure is basically:

My sister + logged out too early + and + missed the last message

The subject min søster applies to both verbs. Danish often does this, just like English:

• Min søster loggede ud for tidligt og gik glip af den sidste besked.

You could also repeat the subject, but it is not necessary:

• Min søster loggede ud for tidligt, og hun gik glip af den sidste besked.

Yes, missede is possible in modern Danish, especially in informal speech, because at misse is commonly used.

So both can work:

• Hun missede den sidste besked
• Hun gik glip af den sidste besked

But gå glip af is a very common native Danish expression and is often the more idiomatic choice.

Be careful not to confuse this with miste, which usually means lose, not miss in this context:

• Jeg mistede min telefon = I lost my phone
• Jeg gik glip af beskeden = I missed the message