Min serviet falder på gulvet, så hun giver mig en ny.

Because serviet is a common gender noun in Danish: en serviet.

Possessive words change to match the noun:

• min for common gender singular
• mit for neuter singular
• mine for plural

So:

• min serviet = correct
• mit serviet = not correct

That -r is the normal present tense ending in Danish.

So:

• at falde = to fall

• falder = falls / is falling

• at give = to give
• giver = gives / is giving

A very important difference from English: Danish verbs do not change according to the subject.

So you get:

• jeg falder
• du falder
• hun falder
• vi falder

The same is true for giver.

På means on here, and gulvet means the floor.

A few useful points:

• gulv is a neuter noun: et gulv
• the definite form is gulvet = the floor

So:

• på gulvet = on the floor / onto the floor

With falde, Danish normally uses på for this idea:

• Bogen falder på gulvet = The book falls on the floor

English sometimes distinguishes on and onto more clearly, but Danish often just uses på.

Because the sentence is talking about the floor in the situation you are already in — the floor of the room, restaurant, etc. That makes the floor the natural choice.

So:

• på gulvet = on the floor
• på et gulv = on a floor

På et gulv would sound more general or less specific, as if you were talking about any floor rather than the actual one in the scene.

Here så means so or therefore.

It connects the two ideas like this:

• My napkin falls on the floor, so she gives me a new one.

So it shows a result:

1. the napkin falls
2. as a result, she gives me another one

Because this så is being used as a coordinating conjunction meaning so.

After a coordinating conjunction, the next clause keeps normal main-clause word order:

• hun giver mig en ny

So:

• Min serviet falder på gulvet, så hun giver mig en ny.

If så meant something more like then at the beginning of a clause, you would often get inversion:

• Så giver hun mig en ny. = Then she gives me a new one.

So the two patterns are related, but not identical.

Because mig is the object form of jeg.

Compare:

• jeg = I
• mig = me

In this sentence, hun is the subject, and mig is the person receiving the thing:

• hun giver mig en ny = she gives me a new one

So jeg would be wrong here.

Because Danish usually follows the pattern give someone something directly:

• hun giver mig en ny
• literally: she gives me a new one

This is the normal, neutral structure.

You can sometimes use til in other contexts, but with give plus a recipient, Danish very often just uses the indirect object directly:

• give mig bogen
• give hende en serviet

So giver mig en ny is the most natural wording here.

En ny means a new one, with serviet understood from the context.

Danish often leaves out a noun if it is obvious:

• Jeg tager en kaffe. Vil du også have en?
• Min serviet falder på gulvet, så hun giver mig en ny.

In the second example, en ny really means:

• en ny serviet

But since that is already clear, Danish does not need to repeat serviet.

Because the missing noun is serviet, and serviet is a common gender noun.

Adjectives and articles must match that noun:

• en ny serviet = a new napkin
• et nyt glas = a new glass

So when the noun is left out, the agreement still stays:

• en ny = a new one, referring to en serviet
• et nyt = a new one, referring to a neuter noun

That is why en ny is correct here.

Yes, but the most neutral version here is min serviet.

Both are possible:

• min serviet
• servietten min

Very roughly:

• min serviet is the standard, straightforward way to say my napkin
• servietten min can sound more specific, contrastive, or slightly more colloquial in some contexts

So in a simple sentence like this, min serviet is the safest and most natural choice.