Плата за вход дорогая.

Because it agrees with плата, which is a feminine noun.

• плата = fee / payment / charge
• feminine singular nominative noun
• so the adjective must also be feminine singular nominative: дорогая

Compare:

• Плата дорогая. = The fee is expensive.
• Вход дорогой. = Entry is expensive. / The entrance ticket is expensive.
  Here дорогой is masculine because вход is masculine.

Because за in this expression means for, and with this meaning it normally takes the accusative case.

So:

• за вход = for entry / for admission

The noun вход is masculine and inanimate. For masculine inanimate nouns, the accusative singular looks the same as the nominative singular. That is why it stays вход.

Compare:

• nominative: вход
• accusative: вход

So плата за вход literally means a fee for entry.

Not exactly.

• плата = a fee, charge, payment
• цена = price

In this sentence, плата suggests a charge someone must pay, especially for some service or access:

• плата за вход = admission fee / entry fee

You could sometimes say цена билета if you mean the ticket price, but плата за вход focuses more on the fact that there is a charge for entering.

In Russian, the verb to be is usually omitted in the present tense.

So:

• Плата за вход дорогая. literally: Fee for entry expensive. natural English: The entrance fee is expensive.

This is completely normal in Russian.

If you wanted past or future, a form of быть would appear:

• Плата за вход была дорогая. = The entrance fee was expensive.
• Плата за вход будет дорогая. = The entrance fee will be expensive.

Yes, Russian word order is flexible.

The most neutral order here is:

• Плата за вход дорогая.

But you could also say:

• Дорогая плата за вход.
• За вход плата дорогая.

These are possible, but they sound different in emphasis.

Usually:

• Плата за вход дорогая sounds neutral and natural.
• putting дорогая first may sound more expressive or literary
• starting with за вход can emphasize the idea of for entry

For learners, the given order is the safest standard version.

Yes, and that is actually very common.

• Вход дорогой. = Entry is expensive. / Admission is expensive.
• Плата за вход дорогая. = The fee for entry is expensive.

The second version is more explicit because it names the fee itself.
The first version is shorter and often more idiomatic in everyday speech.

So both can work, but:

• Вход дорогой = more natural in many situations
• Плата за вход дорогая = slightly more formal or descriptive

No, not in this sentence.

You need дорогая because it is an adjective describing the noun плата.

• плата = feminine noun
• adjective must agree: дорогая

Дорого is an adverb, not an adjective. It is used in sentences like:

• Здесь дорого. = It’s expensive here.
• Это стоит дорого. = This costs a lot.

But with a noun, use the adjective:

• дорогая плата
• дорогой билет
• дорогое вино

Because платёж and плата are related but not the same.

• плата = fee / charge
• платёж = payment, usually the act or instance of paying

So:

• плата за вход = the admission fee itself
• платёж за вход would sound more like the payment for entry, meaning the act of paying, not the fee as a price

In this sentence, плата is the natural word.

Yes, it is a very common pattern.

Russian often uses:

• плата за
  • accusative
• цена за is less common in the same way; usually цена чего?

Examples:

• плата за проезд = fare / payment for travel
• плата за обучение = tuition / payment for education
• плата за парковку = parking fee
• плата за вход = entry fee

So this is a useful pattern to remember.

In this sentence, it clearly means expensive.

The adjective дорогой / дорогая / дорогое can mean:

1. expensive
2. dear in the emotional sense

But context decides the meaning.

Here, because it refers to a fee, the meaning is definitely:

• дорогая = expensive

No Russian speaker would understand it as beloved or dear here.

Russian has no articles, so Плата за вход дорогая can mean:

• The entrance fee is expensive
• An entrance fee is expensive
• Entrance fee is expensive

In real context, English usually chooses the:

• The entrance fee is expensive.

If Russian wants to be more specific, it can use other words:

• Эта плата за вход дорогая. = This entrance fee is expensive.
• Плата за вход в музей дорогая. = The entrance fee to the museum is expensive.

But there is no direct equivalent of the or a in ordinary Russian grammar.

Grammatically yes, but it is less natural as a full sentence by itself.

• Дорогая плата за вход sounds more like a noun phrase: an expensive entrance fee
• Плата за вход дорогая is a complete statement: The entrance fee is expensive

So if you want to describe something inside a bigger sentence, pre-nominal position is fine:

• Это дорогая плата за вход. = This is an expensive entrance fee.

But as a simple statement, Russian usually prefers:

• Плата за вход дорогая.

Yes, it is grammatical and understandable.

However, in everyday speech many native speakers would more often say:

• Вход дорогой.
• Плата за вход слишком высокая. = The entrance fee is too high.
• Входная плата высокая. = The admission fee is high.

So Плата за вход дорогая is correct, but depending on context, another version may sound more idiomatic. A learner should understand it easily and can also use it, but it is good to know the shorter and more common alternatives too.