Denne regel er vigtig for vores sikkerhed i trafikken.

Denne means this and is used before a singular common-gender noun. Regel is a common-gender noun, so denne regel is correct.

The matching forms are:

• denne = singular common gender
• dette = singular neuter
• disse = plural

So you get denne regel, but for a neuter noun you would say dette problem.

Because the subject regel is a singular common-gender noun. In Danish, a predicative adjective after er often agrees with the noun in gender and number:

• en regel er vigtig
• et problem er vigtigt
• regler er vigtige

So vigtig is the correct form for denne regel.

In Danish, a possessive like vores already determines the noun, so you do not add an article as well. That is why it is:

• vores sikkerhed = our safety

not something like en vores sikkerhed.

This is the same pattern as:

• min bil = my car
• hendes bog = her book

After a possessive, the noun is normally in the indefinite form in Danish. So you say:

• vores sikkerhed
• min bil
• deres hus

not vores sikkerheden, min bilen, or deres huset.

This is a very important pattern in Danish: possessive + noun usually means no article and no definite ending.

Usually no. In modern Danish, vores is the normal form for our with all nouns:

• vores regel
• vores hus
• vores regler

There are older or more formal forms like vor, vort, and vore, but in everyday modern Danish, vores is the standard choice.

Because vigtig commonly takes the preposition for when you say what something is important for or to. So:

• vigtig for vores sikkerhed = important for our safety

This is a normal Danish pattern, just like:

• god for helbredet = good for health
• nødvendig for arbejdet = necessary for the work

So here for is simply the natural preposition that goes with vigtig.

Because Danish uses i for being in a situation, area, or environment like traffic. I trafikken means something like in traffic or on the road.

Using på here would sound wrong. Danish says:

• i trafikken
• i skolen
• i byen

So this is mainly a matter of normal Danish preposition usage.

In Danish, the definite form is often used for a general area or sphere of activity. I trafikken does not mean one specific traffic jam; it means in traffic / in road traffic in a general sense.

So even though English might speak more generally, Danish often prefers the definite form in expressions like this. Here trafikken is best understood as the traffic environment or road traffic as a whole.

Yes. Den her regel is very common in spoken Danish and often sounds more natural in everyday conversation. Denne regel is also correct, but it can sound a bit more formal, written, or emphatic.

So the difference is mostly one of style:

• denne regel = correct, often a bit more formal
• den her regel = very common in everyday speech

The sentence follows normal Danish main-clause word order:

Denne regel + er + vigtig for vores sikkerhed i trafikken

Danish is a V2 language, which means the finite verb normally comes in the second position in a main clause. Here:

• first position: Denne regel
• second position: er

If you move another element to the front, the verb still stays second:

• I trafikken er denne regel vigtig for vores sikkerhed.

That is a very typical Danish pattern.