Bez routeru připojení k internetu nefunguje.

Because bez always takes the genitive case in Czech.

• router = the basic dictionary form, nominative
• bez routeru = without a router

So this is a very common pattern:

• bez cukru = without sugar
• bez auta = without a car
• bez problémů = without problems

Here, router is a masculine inanimate noun, and in the singular genitive it becomes routeru.

It is in the dative case, because of the preposition k.

• k means to, toward, or in expressions like this often for / to
• internet → internetu in the singular dative

So:

• připojení k internetu = connection to the internet

This is a fixed and very common combination in Czech:

• přístup k informacím = access to information
• klíč k bytu = key to the apartment
• cesta k úspěchu = road to success

No, here připojení is a noun, not a verb.

It means connection.

It comes from the verb připojit (se) = to connect, but in this sentence it functions as a noun:

• připojení = connection
• internetové připojení = internet connection
• připojení k internetu = connection to the internet

So the sentence structure is basically:

• Bez routeru = without a router
• připojení k internetu = the internet connection
• nefunguje = does not work

Because the subject is singular: připojení.

In Czech, the verb agrees with the subject in number:

• připojení nefunguje = the connection does not work
• připojení nefungují = the connections do not work

Even though the phrase k internetu follows it, that is just part of the noun phrase and does not affect the verb agreement.

Because Czech has no articles.

English distinguishes between:

• a router
• the router

Czech usually just says:

• router

The exact meaning depends on context.

So Bez routeru připojení k internetu nefunguje could mean:

• Without a router, the internet connection does not work.
• Without the router, the internet connection does not work.

Czech leaves that unspecified unless the context makes it clear.

Yes. Czech word order is more flexible than English word order.

The given sentence is perfectly natural:

• Bez routeru připojení k internetu nefunguje.

But you could also hear:

• Připojení k internetu bez routeru nefunguje.

Both mean essentially the same thing. The difference is mostly one of emphasis or information structure.

• Starting with Bez routeru emphasizes the condition first: without a router
• Starting with Připojení k internetu puts the topic first: the internet connection

In English, changing the order often sounds strange, but in Czech it is much more normal.

Because nefunguje is already a complete finite verb meaning does not work / is not working / doesn’t function.

You do not need je with it.

Compare:

• To nefunguje. = That doesn’t work.
• Připojení nefunguje. = The connection doesn’t work.

You would use je with a different kind of predicate, for example:

• Připojení je pomalé. = The connection is slow.

But with fungovat, the verb itself carries the meaning.

Yes, literally it means functions, but in normal English we usually translate it as works.

So:

• funguje = functions / works
• nefunguje = does not function / doesn’t work

In everyday Czech, fungovat is extremely common and often matches English work:

• Počítač nefunguje. = The computer doesn’t work.
• Wi‑Fi nefunguje. = The Wi‑Fi isn’t working.
• To nebude fungovat. = That won’t work.

It is a borrowed word, but yes, it is commonly used in Czech.

Czech often borrows modern technical words from English and then declines them according to Czech grammar. So even though router comes from English, it behaves like a Czech noun in sentences:

• router = nominative
• bez routeru = genitive
• k routeru = dative
• s routerem = instrumental

So learners should get used to this pattern: borrowed nouns often still take normal Czech case endings.

In standard Czech, bez routeru is the safer and more natural form.

Some masculine inanimate nouns can have more than one genitive ending in different varieties of Czech, but with router, routeru is the standard form you should use.

So for learners:

• use bez routeru
• avoid bez routera unless you are specifically dealing with nonstandard or regional usage

Yes, very much so. It is a standard Czech way to say internet connection or connection to the internet.

You may also see:

• internetové připojení = internet connection
• připojení na internet = connection to the internet

But připojení k internetu is completely normal and common.

The prepositions can vary a bit depending on style and nuance, but for a learner, připojení k internetu is an excellent phrase to remember.

A simple way to understand it is:

• Bez routeru = adverbial phrase expressing condition/circumstance
• připojení k internetu = subject
• nefunguje = verb

So literally the structure is:

Without a router, connection to the internet does not work.

Czech does not need a dummy subject like English it, and it also does not need articles. That is why the sentence can look shorter and more compact than the English equivalent.