kono kouen ha eki kara tooi desu ga, sizuka desu.

Yes. The grammar is fine, and it sounds natural in polite Japanese.

In normal writing, though, Japanese is usually written without spaces:

この公園は駅から遠いですが、静かです。

A learner version may add spaces to make the words easier to see, but native writing normally does not.

この must be used before a noun, so この公園 means this park.

By contrast, これ stands by itself and means this one. So:

• この公園 = this park
• これ = this one

You cannot normally say これ公園.

Because この directly modifies a noun by itself. It does not need の.

So these are correct:

• この公園
• その駅
• あの人

Using の there would be incorrect.

は marks the topic of the sentence. It tells you what the sentence is talking about.

So この公園は... is like saying:

As for this park, ...

The rest of the sentence then describes that park.

This is one of the most common sentence patterns in Japanese:

• A は B です
• As for A, it is B

から marks the starting point or reference point. With distance expressions, Japanese often uses X から 遠い / 近い to mean far from X / close to X.

So:

• 駅から遠い = far from the station
• 駅から近い = close to the station

Using に here would not be the normal choice.

Because they are two different types of Japanese adjectives.

• 遠い is an い-adjective
• 静か is a な-adjective

That affects how they behave.

Examples:

• 遠い公園 = a faraway park
• 静かな公園 = a quiet park

And in polite predicate form:

• 遠いです
• 静かです

So the difference is grammatical, not random.

Because 遠い is an い-adjective, and the い is part of the adjective itself.

So you keep it:

• 遠いです ✔
• 遠です ✘

This is true for most い-adjectives:

• 高いです
• 安いです
• 新しいです

You do not remove the final い when adding です.

Because 静か is a な-adjective.

For a な-adjective:

• before a noun: use な
  静かな公園
• at the end of a sentence: use です
  公園は静かです

So:

• 静かです ✔
• 静かいです ✘
• 静かなです ✘

The な only appears when the adjective directly modifies a noun.

Here, が is not the subject particle. It is a conjunction meaning something like but or although.

So:

駅から遠いですが、静かです。

means the first clause contrasts with the second:

• it is far from the station,
• but it is quiet.

Japanese が in this use is often a bit softer than a strong English but. It can sound like a mild contrast.

Because both parts of the sentence are being stated politely.

• 遠いです
• 静かです

That keeps the whole sentence in a consistent polite style.

If you switched to plain style, it would look different, for example:

この公園は駅から遠いが、静かだ。

But in polite everyday Japanese, 遠いですが、静かです is completely normal.

Yes. けど is very common and more conversational.

For example:

この公園は駅から遠いけど、静かです。

Both are natural, but the nuance is slightly different:

• が = a bit more formal or neutral
• けど = a bit more casual/conversational

So the sentence with が is a good polite textbook-style sentence.