Потайная дверь открылась.

Because дверь is the subject of the sentence: it is the thing that opened.

In Russian, the subject is normally in the nominative case, just like in English we would say the door opened.

• дверь открылась = the door opened
• открыть дверь = to open the door

That second example is different because дверь is the object there, not the subject.

Because it has to agree with дверь.

Russian adjectives change their endings to match the noun in:

• gender
• number
• case

Here, дверь is:

• feminine
• singular
• nominative

So the adjective also takes the feminine singular nominative ending:

• потайный = masculine dictionary form
• потайная = feminine singular nominative

So:

• потайная дверь = hidden/secret door

Because the verb is in the past tense, and in Russian past tense verbs agree with the subject in gender and number.

Since дверь is feminine singular, the past tense form is feminine too:

• открылся = masculine
• открылась = feminine
• открылось = neuter
• открылись = plural

Compare:

• стол открылся = the table opened
• дверь открылась = the door opened
• окно открылось = the window opened
• двери открылись = the doors opened

The -сь is the reflexive marker. After a vowel, Russian usually uses -сь instead of -ся.

The basic idea here is that открыться means to open in the sense of to become open.

So:

• открыть = to open something
• открыться = to open / to become open

Compare:

• Он открыл дверь. = He opened the door.
• Дверь открылась. = The door opened.

In many cases, reflexive verbs in Russian are not literally translated with self in English. So открылась does not sound like opened itself in normal English, even though historically that is the idea behind the form.

Открылась is the perfective past form of открыться. Perfective aspect presents the action as a completed event.

So дверь открылась means the door did open / came open.

If you use the imperfective открывалась, the meaning changes:

• дверь открылась = the door opened, as a completed event
• дверь открывалась = the door was opening / used to open / would open

The imperfective focuses more on the process, repetition, or background description.

Examples:

• Когда он нажал кнопку, дверь открылась.
  When he pressed the button, the door opened.

• Дверь медленно открывалась.
  The door was slowly opening.

Not exactly.

Both can relate to the idea of secret or hidden, but потайная often describes something concealed physically or built to be hidden.

So потайная дверь is very natural for:

• a hidden door in a wall
• a concealed panel
• a secret passage entrance

Тайная дверь is possible in some contexts, but потайная дверь sounds more specifically like a hidden/concealed door.

Russian has no articles.

So дверь can mean:

• a door
• the door

Which one is intended depends on context.

That means Потайная дверь открылась could mean:

• A hidden door opened
• The hidden door opened

Usually the broader context makes it clear.

Yes. Russian word order is more flexible than English because case endings and verb forms carry a lot of grammatical information.

The neutral order here is:

• Потайная дверь открылась.

But other orders are possible, depending on emphasis:

• Открылась потайная дверь.
  This can sound more dramatic, like A hidden door opened.

• Дверь потайная открылась.
  Possible in special contexts, but less neutral.

So the original order is the most straightforward and natural for a basic statement.

Good question. Not all feminine nouns end in -а or -я.

Дверь is one of many feminine nouns ending in a soft sign (ь). You usually have to learn the gender of these nouns individually, though some patterns help.

Because дверь is feminine, it controls:

• потайная and not потайный
• открылась and not открылся

So in this sentence, both the adjective and the verb confirm that дверь is feminine.

The dictionary form is открыться.

That is the infinitive, meaning to open or to become open.

How the form breaks down:

• откры- = verb stem
• -л- = past tense marker
• -а = feminine singular past ending
• -сь = reflexive marker

So:

• открыться = infinitive
• открылся / открылась / открылось / открылись = past tense forms

That would usually require a different structure, because now the door is not acting as the subject of an intransitive verb; it is receiving an action.

For example:

• Потайную дверь открыли. = They opened the hidden door.
• Потайная дверь была открыта. = The hidden door was opened / was open, depending on context

So:

• Потайная дверь открылась. = The hidden door opened.
• Потайную дверь открыли. = Someone opened the hidden door.

This is an important difference in Russian:

• открыться = to open
• открыть что-то = to open something

Yes, it is completely natural.

It sounds like a normal sentence you might see in:

• fiction
• fantasy
• detective stories
• games
• descriptions of hidden mechanisms

It is short, idiomatic, and grammatically straightforward. It also has a slightly dramatic literary feel, which fits the idea of a hidden door opening.