Learn Hardcore Czech: V nemocnici se nesmí kouřit kvůli zdraví nemocných lidí. - In the hospital it is not allowed to smoke because of the health of ill people.

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.

What exactly does nesmí mean? Is it “must not” or “cannot”? Why not nemůže?

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.

Why is kouřit in the infinitive? Could it be something like kouří instead?

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).

What case is v nemocnici and why does nemocnice change to nemocnici?

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

Could I say Na nemocnici se nesmí kouřit? What is the difference between v and na here?

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.

What does kvůli mean, and what case does it take? What about zdraví?

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.

Could I use protože instead of kvůli? For example: V nemocnici se nesmí kouřit, protože zdraví nemocných lidí.

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.

Why is it zdraví nemocných lidí? What case is nemocných lidí and how does it work?

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.

Could I just say kvůli zdraví nemocných and leave out lidí?

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).

What is the difference between nemocní lidé, nemocní, pacienti, and lidé here?

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.

What is the difference between lidé and lidi? Why is it lidí here?

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.

Could I say V nemocnici je zakázáno kouřit kvůli zdraví nemocných lidí instead of V nemocnici se nesmí kouřit …? Is there a difference?

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.”

Could I say V nemocnici se nekouří kvůli zdraví nemocných lidí? How does that differ from se nesmí kouřit?

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.