Das Fenster ist geöffnet.

Because Fenster is a neuter noun in German, and the singular nominative article for neuter nouns is das.

• der = masculine
• die = feminine
• das = neuter

So the dictionary form is das Fenster.

Note that die Fenster does exist, but that would be plural: the windows.

It is in the nominative case because it is the subject of the sentence.

A quick way to see that:

• What is opened / is open? → Das Fenster

So das Fenster is the thing the sentence is about, which makes it the subject.

Because öffnet and ist geöffnet mean different things grammatically.

• öffnet = opens / is opening depending on context
  Example: Das Fenster öffnet sich. = The window opens / is opening.
• ist geöffnet = is open / is opened in the sense of a state

So Das Fenster ist geöffnet describes the condition of the window, not the action of opening it.

It is originally the past participle of öffnen (to open), but in this sentence it functions very much like an adjective after sein.

So in Das Fenster ist geöffnet, geöffnet describes the state of the window.

This is why learners often understand it as:

• literally: The window is opened
• naturally: The window is open

It is best understood as a state passive (often called Zustandspassiv in German grammar).

German often distinguishes between:

• Das Fenster wird geöffnet.
  = The window is being opened.
  This focuses on the action/process.

• Das Fenster ist geöffnet.
  = The window is open / has been opened.
  This focuses on the resulting state.

So yes, it is related to the passive, but more specifically it describes the state after the action.

Both can often be translated as The window is open, but there is a small nuance.

• Das Fenster ist offen
  focuses on the simple fact that it is open.

• Das Fenster ist geöffnet
  can sound a bit more formal and may suggest that it is in the state of having been opened.

In everyday speech, offen is often more common for physical things like doors and windows:

• Die Tür ist offen.
• Das Fenster ist offen.

But geöffnet is absolutely correct.

Because the sentence uses the verb sein (to be) to describe a state.

• ist geöffnet = is open / is opened
• hat geöffnet would mean something completely different and usually would not work with Fenster in this way.

German uses sein + participle here because it is describing what the window is like now, not what it has done.

Because it comes after the verb ist and is being used predicatively.

In German, adjectives used this way do not take adjective endings.

Compare:

• Das Fenster ist geöffnet.
  No ending, because geöffnet comes after ist.

• das geöffnete Fenster
  Here it comes before the noun, so it does take an ending: -e.

This is the same pattern as:

• Der Mann ist alt.
• der alte Mann

Yes, but that is not a complete sentence by itself. It means:

• the opened/open window

This is a noun phrase, not a full statement.

Examples:

• Das geöffnete Fenster ist kalt. = The open window is cold.
• Ich sehe das geöffnete Fenster. = I see the open window.

So:

• Das Fenster ist geöffnet. = full sentence
• das geöffnete Fenster = noun phrase

It comes from the verb öffnen (to open).

The past participle is formed as:

• öffnen → geöffnet

This follows a common pattern for regular verbs:

• ge-
  • verb stem + -t / -et

Because the stem ends in a sound that works better with an extra syllable, German uses -et here:

• öffn- → geöffnet

Other examples:

• lernen → gelernt
• machen → gemacht
• öffnen → geöffnet

It is grammatically in the present tense because of ist.

However, the participle geöffnet refers to a state that usually results from a previous action. So the idea is:

• someone opened the window earlier
• now the window is open

That is why English may translate it naturally as The window is open, even though German uses the participle geöffnet.

This is standard German main-clause word order:

• Das Fenster = subject
• ist = finite verb in second position
• geöffnet = predicate at the end

So the pattern is:

Subject + verb + complement

This is one of the most basic German sentence patterns.

Yes. geöffnet is often used in formal or official contexts, especially for businesses, institutions, or times of operation:

• Das Geschäft ist geöffnet. = The shop is open.
• Die Ausstellung ist geöffnet. = The exhibition is open.

For a window, offen is often more everyday and natural, but geöffnet is still fully correct.