V nemocnici se nesmí kouřit kvůli zdraví nemocných lidí.

Se nesmí kouřit is an impersonal / general construction that means “it is not allowed to smoke” or “one must not smoke.”

• se here does not mean himself/herself/itself.
• It is a so‑called reflexive passive / general construction used to express general rules:
  • Tady se nekouří. – People don’t smoke here / No smoking here.
  • V kině se nesmí mluvit. – You must not talk in the cinema.

If you dropped se and said V nemocnici nesmí kouřit, it would sound like an incomplete sentence in Czech: “In the hospital must not smoke … (who?)”
You’d need a subject:

• On nesmí kouřit v nemocnici. – He is not allowed to smoke in the hospital.

For an impersonal rule, Czech very often uses se + 3rd person singular like in se nesmí kouřit.

Nesmí is from smět = to be allowed / may.
So nesmí means “is not allowed / must not / may not.”

• V nemocnici se nesmí kouřit. – Smoking is not allowed in the hospital / You must not smoke in the hospital.

By contrast, nemůže is from moci = can / be able to. It means is not able to or cannot:

• Nemůže kouřit, protože nemá cigarety. – He can’t smoke because he has no cigarettes.

Here we are talking about a rule / prohibition, so nesmí is the correct verb, not nemůže.

With verbs like smět / nesmět (may / may not, be allowed / not allowed), Czech uses the infinitive of the main verb:

• smím kouřit – I may smoke
• nesmí kouřit – he/she/they must not smoke

So nesmí kouřit literally: “(one) may not smoke.”

If you said V nemocnici se nekouří, that is a different structure:

• se nekouří = people do not smoke here / no one smokes here (as a fact, habit).

Nesmí kouřit → focuses on a rule (it is forbidden).
Nekouří → describes what actually happens (no one smokes there).

Nemocnice is a feminine noun.

After v meaning “in” (static location), Czech uses the locative case:

• nominative: nemocnice – a hospital
• locative singular: (o) nemocnici – about the hospital, in the hospital

So:

• V nemocnici = “in (the) hospital” – locative singular.

This is the regular pattern for many feminine nouns ending in -e:

• ulice → v ulici – street → in the street
• restaurace → v restauraci – restaurant → in the restaurant

No, in this sentence na nemocnici would sound wrong or very strange.

• V nemocnici = inside the hospital building / within the hospital.
• Na nemocnici would literally be “on the hospital” (e.g. physically on the roof or on the surface).

For buildings where you mean inside the building, you normally use v:

• v nemocnici – in the hospital
• v kině – in the cinema
• v obchodě – in the shop

There are places/institutions where na is normal (because Czech treats them differently):
na poště (at the post office), na univerzitě (at the university), na nádraží (at the station).
But nemocnice (hospital) normally takes v when you mean being inside it.

Kvůli means roughly “because of / due to / for the sake of.”
It always takes the dative case.

Zdraví (health) is a neuter noun whose nominative and dative singular look the same: zdraví.

So in kvůli zdraví we actually have:

• preposition kvůli
  • dative singular zdraví
    → “because of the health / for the sake of the health.”

More examples with kvůli + dative:

• Kvůli tobě jsem přišel pozdě. – Because of you I came late.
• Musíme odejít kvůli dětem. – We must leave because of the children / for the children’s sake.

You can use protože, but you must make a full clause after it:

• V nemocnici se nesmí kouřit, protože to škodí zdraví nemocných lidí.
  – You must not smoke in the hospital because it harms the health of sick people.

The difference:

• kvůli + noun: because of / for the sake of something
  → kvůli zdraví nemocných lidí – because of the health of sick people
• protože + clause: because
  • full sentence
    → protože to škodí zdraví nemocných lidí – because it harms their health

So your example protože zdraví nemocných lidí is incomplete in Czech. After protože you need a verb.

Zdraví here is a noun: health.

The phrase nemocných lidí is in the genitive plural:

• nemocní lidé – sick people (nominative plural)
• nemocných lidí – of sick people (genitive plural)

Structure:

• zdraví (koho? čeho?) nemocných lidí
  → the health of sick people

In Czech, the relationship “X of Y” is usually expressed by Y in the genitive:

• dům mé sestry – my sister’s house (house of my sister)
• zdraví dětí – children’s health (health of children)
• zdraví nemocných lidí – the health of sick people / patients’ health

So nemocných is an adjective in genitive plural, agreeing with lidí in genitive plural.

Yes, that is very natural and even more typical:

• V nemocnici se nesmí kouřit kvůli zdraví nemocných.

Czech often drops the noun when the meaning is clear from context. Here nemocní clearly means “sick people / patients.”

So:

• nemocní (nominative plural) – the sick (people)
• nemocných (genitive plural) – of the sick (people)

Kvůli zdraví nemocných = for the health of the sick (i.e. patients).

All are possible with slightly different nuance:

• nemocní lidé – sick people (more descriptive, neutral)
• nemocní – the sick (shorter, very common in this context)
• pacienti – patients (more medical / formal)
• lidé – people (general)

In this sentence you could say:

• kvůli zdraví nemocných lidí – because of the health of sick people
• kvůli zdraví nemocných – because of the health of the sick
• kvůli zdraví pacientů – because of the health of the patients

All are correct; kvůli zdraví pacientů and kvůli zdraví nemocných are very natural in a hospital context.

Both lidé and lidi mean “people.”

• lidé – more formal / standard
• lidi – more colloquial, everyday speech

In the genitive plural, both become lidí (same form), so here we see:

• nominative: lidé / lidi – people
• genitive: lidí – of people

So nemocných lidí = “of sick people.”
The choice lidé vs. lidi only shows up in the nominative; in genitive plural both are lidí.

Yes, you can. Both are correct but the style is a bit different:

• V nemocnici se nesmí kouřit
  – very common, everyday way to express a rule/prohibition.

• V nemocnici je zakázáno kouřit
  – slightly more formal / written, sounds like something on an official sign or regulation.

Meaning is almost the same: “Smoking is forbidden / not allowed in the hospital.”

Yes, that sentence is grammatically fine, but the nuance changes:

• V nemocnici se nesmí kouřit
  → focuses on the rule: You must not smoke there; it’s forbidden.

• V nemocnici se nekouří
  → describes the fact / habit: People don’t smoke there; no one smokes there.

Adding kvůli zdraví nemocných lidí explains the reason in both cases, but:

• With se nesmí kouřit, you sound like you are stating the regulation.
• With se nekouří, you sound like you are describing what happens, possibly because of that rule.