Promítačka je za filmovým plátnem.

Because the preposition za changes the case of the noun.

In this sentence, za means behind in the sense of location: the projector is already there, not moving there. With that meaning, za takes the instrumental case.

• dictionary form: plátno = screen / canvas
• instrumental singular: plátnem

So:

• za plátnem = behind the screen

A very useful contrast is:

• Je za plátnem. = It is behind the screen. → location → instrumental
• Jde za plátno. = It is going behind the screen. → movement toward a place → accusative

So plátnem appears because Czech marks location with za + instrumental here.

Because the adjective has to match the noun in gender, number, and case.

The noun plátno is:

• neuter
• singular
• here in the instrumental case

So the adjective filmový must also be neuter singular instrumental:

• filmový → basic form
• filmovým → instrumental singular

That gives:

• filmovým plátnem = by/with/behind a movie screen, depending on context

Both words change together:

• filmové plátno = movie screen
• za filmovým plátnem = behind the movie screen

Here za means behind.

But za is a very flexible Czech preposition, so learners often get confused by it. It can mean different things depending on context and case, for example:

• za domem = behind the house
• za hodinu = in an hour
• děkuji za pomoc = thank you for the help
• platit za kávu = to pay for coffee

In your sentence, za + instrumental expresses position behind something:

• Promítačka je za filmovým plátnem. = The projector is behind the movie screen.

So the key point is not just the word za, but the pattern za + instrumental for location.

Yes, promítačka is a feminine noun.

It refers to a projector or projecting machine. It comes from the verb promítat, meaning to project or to show.

Its ending -a is a common sign of feminine nouns in Czech, and that affects adjectives and pronouns used with it. For example:

• ta promítačka = that projector
• nová promítačka = a new projector

In this sentence, though, the verb je does not change for gender, so you simply get:

• Promítačka je... = The projector is...

In the present tense, the Czech verb být (to be) does not change for gender in the 3rd person singular.

So all of these use je:

• On je doma. = He is at home.
• Ona je doma. = She is at home.
• To je doma. = It is at home.

That is why you have:

• Promítačka je za filmovým plátnem.

Even though promítačka is feminine, the verb is still je.

Gender matters much more for adjectives, participles, and pronouns than for this present-tense form of být.

Yes, absolutely.

Czech word order is much more flexible than English word order. Both of these are grammatical:

• Promítačka je za filmovým plátnem.
• Za filmovým plátnem je promítačka.

The difference is mainly one of focus or emphasis.

• Promítačka je za filmovým plátnem.
  This sounds neutral: you are talking about the projector and saying where it is.

• Za filmovým plátnem je promítačka.
  This puts more emphasis on the location, or it may sound like you are answering What is behind the screen?

So English needs stricter word order, but Czech often uses word order to highlight what is new or important in the sentence.

Because Czech has no articles like English a/an and the.

So Czech often leaves that idea to context. The sentence:

• Promítačka je za filmovým plátnem.

could mean:

• The projector is behind the movie screen.
• A projector is behind a movie screen.

In real use, context normally makes it clear which one is meant.

If Czech speakers want to be more specific, they can use things like:

• ta promítačka = that/the projector
• to filmové plátno = that/the movie screen

But usually simple context is enough.

Yes, it is a very natural expression.

• plátno originally means canvas
• in cinema-related contexts, it also means screen
• filmové plátno literally means film/movie screen

So za filmovým plátnem is a very natural way to say behind the movie screen.

In many contexts, Czech speakers could also just say:

• za plátnem = behind the screen

The adjective filmovým just makes it more specific.

One clue is the ending -o, which is very common for neuter singular nouns in Czech.

So:

• plátno → likely neuter
• nominative singular: plátno
• instrumental singular: plátnem

This is a common neuter pattern:

• město → městem
• auto → autem
• plátno → plátnem

So once you recognize plátno as a neuter noun, the form plátnem becomes much easier to understand.

A simple learner-friendly pronunciation guide would be:

proh-MEE-tach-ka yeh za FIL-moh-veem PLAAT-nem

A few key points:

• í is a long vowel
• á is also a long vowel
• č sounds like ch in church
• j in Czech sounds like English y in yes
• ý is a long ee sound in normal pronunciation for most learners

So:

• Promítačka ≈ proh-MEE-tach-ka
• je ≈ yeh
• filmovým ≈ FIL-moh-veem
• plátnem ≈ PLAAT-nem

Yes, you can.

• Promítačka je za plátnem. = The projector is behind the screen.
• Promítačka je za filmovým plátnem. = The projector is behind the movie screen.

The shorter version is perfectly grammatical and natural if the context already makes it clear what kind of screen you mean.

The longer version is just more specific.

So for learning purposes, the sentence with filmovým plátnem is especially useful because it shows how both the adjective and the noun change in the instrumental case.