De beveiliging laat niemand zonder pasje door.

Here de beveiliging usually means the security staff / security team, not the technology itself.

Dutch often uses beveiliging as a collective noun, much like English security:

• De beveiliging staat bij de ingang. = The security staff are at the entrance.
• De beveiliging controleert de bezoekers. = Security checks the visitors.

So even though it refers to people, it is grammatically singular in Dutch.

The full verb is doorlaten, which is a separable verb.

In a main clause, Dutch usually splits separable verbs:

• De beveiliging laat niemand door.

Here:

• laat = the finite verb part
• door = the separable particle, which moves to the end

Compare:

• infinitive: doorlaten
• main clause: laat ... door
• past participle: doorgelaten

This is very common in Dutch:

• opbellen → Ik bel je op.
• meebrengen → Zij brengt koffie mee.

Not really. In this sentence, door is mainly the particle belonging to doorlaten.

So although door often means through, here you should understand iemand doorlaten as one verbal unit meaning something like:

• let someone through
• allow someone to pass
• let someone in/past

That is why door ends up at the end of the clause.

Niemand means no one / nobody and is the most natural choice here.

So:

• De beveiliging laat niemand zonder pasje door. means
• Security lets no one through without a pass.

You could say geen mensen, but that sounds less natural in this sentence and focuses more on people as a group:

• De beveiliging laat geen mensen zonder pasje door.

That is understandable, but niemand is the normal word when English would say nobody / no one.

Dutch main clauses usually follow this pattern:

subject + finite verb + object + other information + separable particle

So here:

• De beveiliging = subject
• laat = finite verb
• niemand = object
• zonder pasje = extra phrase
• door = separable particle

That gives:

• De beveiliging laat niemand zonder pasje door.

This order is very normal and neutral.

Yes, that is possible, but the nuance is a little different.

• De beveiliging laat niemand zonder pasje door.
  This is the most natural neutral version. It clearly says that people who do not have a pass are not allowed through.

• De beveiliging laat niemand door zonder pasje.
  This can also work, but zonder pasje feels a bit more attached to the action at the end, almost like an afterthought or clarification.

In everyday Dutch, the original sentence is the smoother and more standard phrasing.

Pasje is the diminutive form of pas.

• pas = pass, card, badge
• pasje = literally little pass/card

But very often Dutch diminutives do not sound especially small in English. They are just the normal everyday word for certain objects. In this context, pasje is very natural for something like:

• an access card
• an ID badge
• a swipe card

So pasje does not necessarily mean a tiny pass; it is simply idiomatic Dutch.

After zonder (without), Dutch very often leaves out the article when speaking generally.

So:

• zonder pasje = without a pass / without any pass

This is very natural. You can sometimes hear:

• zonder een pasje

but that is more explicit and often less natural in a general statement like this.

Compare:

• zonder jas = without a coat
• zonder geld = without money
• zonder afspraak = without an appointment

Because the subject de beveiliging is grammatically singular.

Even though de beveiliging may refer to several security guards, the noun itself is singular, so the verb must also be singular:

• de beveiliging laat

If the subject were plural, you would use laten:

• De beveiligers laten niemand zonder pasje door.
  = The security guards let no one through without a pass.

Yes, but the meaning shifts slightly.

• de beveiliging = security, the security staff as a group
• de beveiliger = the security guard
• de beveiligers = the security guards

So:

• De beveiliging laat niemand zonder pasje door.
  Security lets no one through without a pass.

• De beveiligers laten niemand zonder pasje door.
  The security guards let no one through without a pass.

The original sentence sounds a bit more general and institutional, which is very natural in announcements or formal situations.