Я живу на седьмом этаже.

Russian normally uses на with этаж when talking about what floor something is on.

• жить на пятом этаже = to live on the fifth floor
• быть на втором этаже = to be on the second floor

So на is the standard preposition here, just like English on in on the seventh floor.

Using в with этаж would sound unnatural in this meaning.

It is the prepositional case.

The preposition на can take different cases, but here it answers the question where? so it uses the prepositional:

• на седьмом этаже = on the seventh floor

Compare:

• где? → на седьмом этаже = where? on the seventh floor
• куда? → на седьмой этаж = to what place? to the seventh floor

So in your sentence, both седьмом and этаже are in the prepositional case.

Because седьмой is an adjective-like word, specifically an ordinal number meaning seventh, and it has to agree with этаж in:

• gender
• number
• case

Since этаж is:

• masculine
• singular
• prepositional

the ordinal also becomes masculine singular prepositional:

• dictionary form: седьмой
• in this sentence: седьмом

So:

• седьмой этаж = the seventh floor
• на седьмом этаже = on the seventh floor

Because it is in the prepositional case after на when meaning where.

• basic form: этаж
• prepositional singular: этаже

This is a normal pattern for many masculine nouns:

• дом → в доме
• сад → в саду / в саде depending on the word
• этаж → на этаже

So этаже simply means on the floor / on the story in the required case form.

Живу is the 1st person singular form of жить:

• жить = to live
• я живу = I live

So:

• Я живу = I live

In this sentence, it means I live / I reside, not I am alive. Russian uses the same verb жить for living somewhere.

Because Russian usually does not use a present-tense form of to be in sentences like this.

English says:

• I am at home
• I am a student

Russian often just says:

• Я дома
• Я студент

In your sentence, the main verb is живу = live, so no extra am is needed.

Yes. Russian word order is more flexible than English.

The neutral order here is:

• Я живу на седьмом этаже.

But you could also hear:

• На седьмом этаже я живу.
• Живу на седьмом этаже.

These versions may shift the emphasis a little:

• Я живу на седьмом этаже = neutral statement
• На седьмом этаже я живу = emphasis on where
• Живу на седьмом этаже = I live on the seventh floor, with I understood from the verb

The stress is:

• Я живу́ на седьмо́м этаже́.

A rough pronunciation guide:

• Я = ya
• живу́ = zhi-VOO
• на = na
• седьмо́м = syd-MOM
• этаже́ = eh-ta-ZHE

A few notes:

• ж sounds like the s in measure
• the final е in этаже́ is pronounced clearly
• дьм in седьмом can feel tricky; listen and repeat slowly at first

Yes, in this context этаж means floor or story of a building.

Examples:

• первый этаж = first floor
• второй этаж = second floor
• на верхнем этаже = on the top floor

Be careful: English floor can also mean the surface you walk on inside a room. Russian usually uses different words depending on the meaning:

• этаж = storey/floor of a building
• пол = floor surface

You change the verb жить.

• Я жил на седьмом этаже. = I lived on the seventh floor.
  
  • if the speaker is male
• Я жила на седьмом этаже. = I lived on the seventh floor.
  
  • if the speaker is female

Future:

• Я буду жить на седьмом этаже. = I will live on the seventh floor.

The phrase на седьмом этаже stays the same.

Usually, yes in the literal sense of numbered floors, but there can be a cultural difference depending on what kind of English you speak.

In Russian usage:

• первый этаж is the level at ground level

So if you are used to American English, this matches first floor.

If you are used to British English, note that Russian первый этаж corresponds to British ground floor, and Russian второй этаж corresponds to British first floor.

So на седьмом этаже may refer to a different numbered level than a British learner first expects.

Yes. Russian often drops subject pronouns when the verb already makes the subject clear.

• Живу на седьмом этаже. = I live on the seventh floor.

Because живу already means I live, я is optional unless you want emphasis or contrast:

• Я живу на седьмом этаже, а он — на восьмом. = I live on the seventh floor, and he lives on the eighth.