Hvis tallerkenen er beskidt, vasker jeg den med det samme.

In Danish, the definite article is usually attached to the end of the noun.

• en tallerken = a plate
• tallerkenen = the plate

So -en here is the definite ending for a common-gender noun.

This is very common in Danish:

• en bil → bilen
• en kop → koppen
• en tallerken → tallerkenen

Use hvis for a condition: if.

• Hvis tallerkenen er beskidt ... = If the plate is dirty ...

You generally do not use om for this kind of if. Om is usually used for whether:

• Jeg ved ikke, om tallerkenen er beskidt.
  = I don’t know whether the plate is dirty.

So:

• hvis = if in a condition
• om = whether

This is because Danish main clauses follow the verb-second rule.

The first position in the sentence is taken by the whole if-clause:

• Hvis tallerkenen er beskidt

After that, the finite verb of the main clause must come second:

• vasker

Then comes the subject:

• jeg

So the structure is:

• Hvis tallerkenen er beskidt, vasker jeg den med det samme.

If you put the main clause first, you get normal subject-verb order:

• Jeg vasker den med det samme, hvis tallerkenen er beskidt.

Both are correct, but when the sentence starts with the subordinate clause, Danish requires vasker jeg.

Because tallerken is a common-gender noun, not a neuter noun.

In Danish:

• en-word nouns usually take den
• et-word nouns usually take det

Since it is:

• en tallerken

you refer back to it with:

• den

Examples:

• Jeg ser tallerkenen. Jeg vasker den.
• Jeg ser bordet. Jeg vasker det.

So den matches the gender of tallerken.

Here, beskidt is a predicative adjective, because it comes after er:

• tallerkenen er beskidt

Predicative adjectives can show agreement in Danish, but beskidt already ends in -t, so the singular forms look the same.

Compare:

• en tallerken er beskidt
• et gulv er beskidt
• tallerkenerne er beskidte

So in the singular, beskidt stays beskidt here.

You also see it before a noun:

• en beskidt tallerken
• et beskidt gulv
• beskidte tallerkener

Med det samme is an idiomatic expression meaning:

• immediately
• right away
• at once

So:

• vasker jeg den med det samme
  = I wash it immediately / right away

It is best learned as a fixed phrase. Its literal parts are less important than its overall meaning.

A similar word is:

• straks = immediately

So you could also say:

• Hvis tallerkenen er beskidt, vasker jeg den straks.

Danish often uses the present tense for:

• habits
• general rules
• things that happen whenever a condition is true

So this sentence sounds like a general pattern:

• If the plate is dirty, I wash it right away.

That is very natural in Danish.

If you want to stress a future decision, you could say:

• Hvis tallerkenen er beskidt, vil jeg vaske den med det samme.

But in many everyday cases, simple present is preferred and sounds more natural.

Because the sentence begins with a subordinate clause:

• Hvis tallerkenen er beskidt

and then moves to the main clause:

• vasker jeg den med det samme

Danish normally places a comma between those two clauses:

• Hvis tallerkenen er beskidt, vasker jeg den med det samme.

The comma helps show where the subordinate clause ends and the main clause begins.

Yes. That is also correct.

Both of these are grammatical:

• Hvis tallerkenen er beskidt, vasker jeg den med det samme.
• Jeg vasker den med det samme, hvis tallerkenen er beskidt.

The difference is mostly in focus:

• starting with Hvis ... highlights the condition first
• starting with Jeg vasker ... highlights the action first

But remember: when the sentence starts with the if-clause, the main clause must follow the verb-second pattern, which is why you get vasker jeg.