După film, Anei îi este somn și se culcă.

Because Anei is the dative form of Ana.

In Romanian, some states and feelings are expressed with a structure like:

someone-DATIVE + îi/mi/ți etc. + este + noun

So:

• Anei îi este somn = literally To Ana is sleep
• natural English: Ana is sleepy

This is a very common Romanian pattern for physical or emotional states.

This is a normal feature of Romanian called clitic doubling.

• Anei names the person explicitly.
• îi repeats that person in a short pronoun form.

So Romanian often says both:

• Anei îi este somn
• Mariei îi place muzica
• Băiatului îi este frig

In English, this feels redundant, but in Romanian it is very natural.

If the person is already clear from context, you can leave out the full noun and say just:

• Îi este somn = She/He is sleepy

Literally, it is something like to her is sleep or sleepiness is to her.

But you should learn it as a fixed Romanian expression meaning:

• she is sleepy
• she feels sleepy

Here, somn is a noun meaning sleep, not an adjective.

Romanian often uses this kind of structure where English uses to be + adjective.

It is a noun.

• somn = sleep
• somnoros / somnoroasă = sleepy

So:

• Anei îi este somn = Ana feels sleepy
• Ana este somnoroasă = Ana is sleepy/drowsy

Both can work, but îi este somn is the more idiomatic way to say that someone needs sleep right now.

Yes, you could.

But there is a small difference in feel:

• Anei îi este somn = she feels sleepy right now; this is very idiomatic
• Ana este somnoroasă = she is sleepy/drowsy; more adjectival, more like describing her state

In everyday Romanian, when someone needs sleep at that moment, îi e somn / îi este somn is usually the most natural choice.

Yes. îi este and i-e mean the same thing.

• Anei îi este somn = fuller, more careful style
• Anei i-e somn = shorter, very common in speech and informal writing

Romanian often shortens forms of a fi in everyday language:

• mi-e foame
• ți-e frig
• i-e somn

So the sentence could also naturally appear as:

După film, Anei i-e somn și se culcă.

Se culcă comes from the verb a se culca, which means:

• to go to bed
• to lie down

The se is part of the verb. It is a reflexive marker.

So:

• se culcă = she goes to bed
• literally, it is not best translated word-for-word as herself, even though historically it is reflexive

Many Romanian daily-routine verbs are reflexive, so it is best to learn a se culca as a whole verb.

Because Romanian often drops subject pronouns when the subject is clear.

So instead of saying:

• Ea se culcă

Romanian very often just says:

• Se culcă

In this sentence, the subject is already understood to be Ana, so there is no need to repeat ea.

This is normal in Romanian and much more common than in English.

După means after, so după film means after the movie / after the film.

Romanian does not always match English article usage exactly. In context, film can be understood naturally even without a separate word corresponding neatly to English the.

Also, Romanian definite articles are usually attached to the end of the noun, not written as a separate word:

• film = film / movie
• filmul = the film / the movie

But in expressions like după film, Romanian can sound natural without making the noun explicitly definite in the same way English does.

The given order is natural and neutral:

După film, Anei îi este somn și se culcă.

Romanian word order is fairly flexible, but different orders may sound more marked or place emphasis in different spots.

For example:

• Anei îi este somn după film și se culcă.
  Possible, but less natural as a neutral sentence.

• După film, îi este somn Anei și se culcă.
  Grammatically possible, but this sounds marked or stylistically unusual.

Starting with După film is very natural because it sets the time frame first: After the movie...