Здесь запрещено курить.

Because this sentence is impersonal: it does not have a normal subject like курение or это.

In Russian, forms like разрешено, запрещено, нужно, можно, нельзя are often used in impersonal statements about rules, permission, or necessity. In that kind of sentence, Russian normally uses the neuter singular form, so you get запрещено.

So:

• Здесь запрещено курить. = Smoking is forbidden here.
• not about a specific masculine or feminine noun
• just a general rule

If there were a noun, the form could change:

• Вход запрещён. = Entry is forbidden.
  Here вход is masculine, so запрещён matches it.

Because after words like можно, нельзя, разрешено, запрещено, Russian often uses an infinitive to say what action is allowed, forbidden, necessary, and so on.

So the structure is:

• запрещено + infinitive
• разрешено + infinitive
• нельзя + infinitive
• можно + infinitive

Examples:

• Здесь запрещено курить. = It is forbidden to smoke here.
• Здесь нельзя курить. = You must not smoke here.
• Здесь можно фотографировать. = You may take photos here.

English often uses to + verb or a gerund, but Russian usually uses the plain infinitive.

Because this sentence is talking about the activity of smoking in general, not one single completed act.

Russian usually uses the imperfective infinitive in rules and general prohibitions:

• курить = to smoke, smoking as an activity
• читать = to read
• парковаться = to park

A perfective verb like покурить would sound more like to have a smoke / smoke for a bit, which is much less natural in a general rule.

So for signs and regulations, the imperfective is the normal choice:

• Курить запрещено.
• Здесь нельзя парковаться.

Yes, absolutely. Здесь нельзя курить is very common.

The difference is mainly one of style and nuance:

• Здесь нельзя курить = You can't / mustn't smoke here
• Здесь запрещено курить = Smoking is prohibited here

Запрещено sounds a bit more formal, official, or sign-like.
Нельзя is very common in everyday speech and is often the most natural way to say it conversationally.

So:

• on a sign: Курить запрещено
• in speech: Здесь нельзя курить

Both are correct.

Yes. Russian word order is flexible, and changing it usually changes the focus or emphasis, not the basic meaning.

These are all possible:

• Здесь запрещено курить.
• Курить здесь запрещено.
• Здесь курить запрещено.

All of them mean basically the same thing: smoking is forbidden here.

Typical nuance:

• Здесь запрещено курить. — neutral
• Курить здесь запрещено. — stronger emphasis on smoking
• Здесь курить запрещено. — stronger emphasis on here

For a learner, the original sentence is a very natural neutral pattern.

Yes, you could say Тут запрещено курить.

Both здесь and тут mean here. The difference is mostly stylistic:

• здесь = slightly more neutral or standard
• тут = often a bit more conversational

So:

• Здесь запрещено курить. — neutral, standard
• Тут запрещено курить. — a little more informal

On signs, здесь may feel a bit more natural, but both are possible in everyday Russian.

Not in the same way as a normal finite verb like читает or идут.

In this sentence, запрещено functions as a predicative word meaning something like it is forbidden. For learners, the easiest way to understand it is as part of an impersonal pattern:

• запрещено курить = it is forbidden to smoke
• разрешено войти = it is allowed to enter

Historically and grammatically, it is related to the passive form of запретить / запрещать, but in sentences like this it is best learned as a fixed structure meaning forbidden / prohibited.

Because Russian often omits the present-tense equivalent of to be.

English says:

• It is forbidden to smoke here.

Russian simply says:

• Здесь запрещено курить.

There is no present-tense is in ordinary Russian sentences like this.

Compare:

• Он студент. = He is a student.
• Мне холодно. = I am cold.
• Здесь запрещено курить. = Smoking is forbidden here.

So the absence of is is completely normal.

Very natural. In fact, this is exactly the kind of structure often used on signs.

Common sign-style versions are:

• Курить запрещено.
• Здесь запрещено курить.
• Не курить!

The differences:

• Курить запрещено. — standard formal sign wording
• Здесь запрещено курить. — same idea, with here explicitly included
• Не курить! — shorter, more direct: No smoking! / Do not smoke!

So the sentence you were given sounds normal and idiomatic.

A helpful pronunciation guide is:

Зде́сь запрещено́ кури́ть.

Stress:

• здесь — one syllable, stressed
• запрещено́ — stress on the last syllable
• кури́ть — stress on the last syllable

Roughly:

• здесь ≈ zdyes'
• запрещено́ ≈ zuh-pryee-shee-NO
• кури́ть ≈ koo-REET'

A few pronunciation notes:

• здесь starts with a cluster that can be hard for English speakers
• е after soft consonants often sounds like ye
• the final -ть in курить is soft

Usually, Russian prefers the infinitive in this kind of rule:

• Курить запрещено. — the most natural pattern

A noun like курение can be used, but it tends to sound more formal, abstract, or bureaucratic:

• Курение запрещено. = Smoking is prohibited.

That is also correct, and you may see it on signs. The difference is:

• курить = the action to smoke
• курение = the noun smoking

Both are possible, but with запрещено, the infinitive pattern is extremely common and very useful to learn.