Ved døren ligger der en dørmåtte, så gæsterne kan tørre skoene af, før de går ind.

Because Danish, like German, very often makes compound nouns by joining words together.

• dør = door
• måtte = mat
• dørmåtte = doormat

This is completely normal in Danish. English often writes similar ideas as two words, but Danish usually prefers one compound word.

Other examples:

• hoveddør = front door
• skoletaske = school bag
• kaffekop = coffee cup

Ved means by, near, or at in this kind of context. So ved døren means by the door / at the door.

The noun is døren, which is the definite singular form of dør:

• en dør = a door
• døren = the door

It is definite because the sentence is talking about a specific door, most naturally the entrance door.

Danish often uses so-called position verbs where English would simply use is.

Here, ligger literally means lies. A doormat is something that lies flat on the floor, so Danish naturally says:

• Der ligger en dørmåtte ved døren
  = There is a doormat by the door

This is more idiomatic than using er here. Danish often chooses verbs like:

• ligge = lie
• stå = stand
• sidde = sit

depending on how something is positioned.

This der is a dummy subject, similar to English there in sentences like:

• There is a doormat by the door

So:

• Der ligger en dørmåtte ved døren
  literally = There lies a doormat by the door

It does not mean there as a location here. It is just part of an existential structure: saying that something exists or is present somewhere.

Because Danish main clauses follow the V2 rule: the finite verb normally comes in the second position.

Here, Ved døren has been moved to the front for emphasis or style. Once that happens, the finite verb must come next:

• Ved døren = first position
• ligger = second position
• der = comes after the verb

So:

• Ved døren ligger der en dørmåtte

A more neutral version would be:

• Der ligger en dørmåtte ved døren

Both are correct, but the original sentence starts with the location.

Because the sentence is introducing the doormat as new information.

• en dørmåtte = a doormat
• dørmåtten = the doormat

When something is mentioned for the first time, Danish often uses the indefinite form, just like English does.

If the speaker and listener already knew which mat was being discussed, then dørmåtten could make sense.

Here så means so that.

It introduces a purpose/result idea:

• ... så gæsterne kan tørre skoene af
  = ... so that the guests can wipe/dry their shoes off

So the doormat is there with a practical purpose.

Do not confuse this with other meanings of så, such as:

• så = then
• så = so

The exact meaning depends on context.

Gæsterne is the definite plural form of gæst:

• en gæst = a guest
• gæster = guests
• gæsterne = the guests

It is definite because the sentence means the guests in this situation — the people who are coming in.

In English, the guests and guests are sometimes both possible depending on style, but Danish often uses the definite form where the group is understood from context.

This is very natural in Danish. When talking about things closely connected to people — especially body parts, clothing, and personal items being used in the situation — Danish often uses the definite form instead of a possessive.

So:

• tørre skoene af = wipe/dry off the shoes

means the same as:

• tørre deres sko af = wipe/dry off their shoes

But skoene sounds more idiomatic here.

This is similar to how Danish also often says things like:

• Han vaskede hænderne = He washed his hands
  literally: He washed the hands

Tørre ... af is a particle verb meaning wipe off or dry off.

So:

• tørre skoene af = wipe/dry the shoes off

The word af adds the idea of removing dirt, water, or moisture.

In Danish, with this kind of verb, the object often comes before the particle:

• tørre skoene af
• tage jakken på
• slukke lyset

If the object is a pronoun, this word order is especially common:

• tørre dem af = wipe them off

Because kan is a modal verb meaning can.

After a modal verb in Danish, the next verb stays in the infinitive form:

• kan tørre = can wipe/dry
• kan gå = can walk/go
• kan komme = can come

So:

• gæsterne kan tørre skoene af
  = the guests can wipe their shoes off

This works very much like English.

Because før de går ind is a subordinate clause, and subordinate clauses in Danish have different word order from main clauses.

Main clause:

• De går ind = They go in

Subordinate clause after før:

• før de går ind = before they go in

So the subject de comes before the verb går.

Also, gå ind means go in / go inside:

• går = go / walk
• ind = in, inward

Together, gå ind is a very common expression for entering.

Yes. That would be a very natural alternative.

Compare:

• Der ligger en dørmåtte ved døren...
• Ved døren ligger der en dørmåtte...

The difference is mainly focus:

• Der ligger... is more neutral
• Ved døren... puts the location first, so it highlights where the doormat is

Both are grammatical and idiomatic.