Jeg køber nyt opvaskemiddel i supermarkedet, hvis flasken er tom.

Because opvaskemiddel is a neuter noun: et opvaskemiddel.

In Danish, an adjective in the indefinite singular agrees with the noun’s gender:

• common gender: en ny flaske
• neuter: et nyt opvaskemiddel

So nyt opvaskemiddel is the correct neuter form.

Here opvaskemiddel is being used like a mass noun, similar to English dish soap.

So Danish often says:

• Jeg køber nyt opvaskemiddel = I buy new dish soap

without et.

If you say et nyt opvaskemiddel, it can sound more like a new dishwashing product / a new kind of dish soap / a new brand, rather than just more dish soap in general.

Flasken is the definite form, meaning the bottle.

The sentence assumes there is a specific bottle already known from the situation, probably the bottle of dish soap at home. Danish usually marks definiteness by adding an ending:

• en flaske = a bottle
• flasken = the bottle

So hvis flasken er tom means if the bottle is empty, not just if a bottle is empty.

Hvis means if and introduces a condition.

So hvis flasken er tom means that buying dish soap depends on that condition.

Når means when and often suggests that something is expected to happen, or happens regularly. In some contexts, når could also work, but the nuance changes:

• hvis flasken er tom = if the bottle is empty
• når flasken er tom = when the bottle is empty / whenever the bottle is empty

So hvis is the more neutral conditional choice.

Both are in the present tense:

• køber = buy / am buying
• er = is

Danish often uses the present tense for:

• habits
• general truths
• future actions that are understood from context

So this sentence can mean something like:

• I buy dish soap if the bottle is empty
• or more naturally in English, I’ll buy dish soap if the bottle is empty
• or I buy dish soap whenever the bottle is empty, depending on context

Danish does not need a special future form here.

Yes, and that is very natural:

• Hvis flasken er tom, køber jeg nyt opvaskemiddel i supermarkedet.

When the subordinate clause comes first, the main clause uses inversion:

• køber jeg
• not jeg køber

That is a very important Danish word order rule.

Because hvis flasken er tom is a subordinate clause, and Danish normally separates subordinate clauses with a comma.

So:

• Jeg køber nyt opvaskemiddel i supermarkedet, hvis flasken er tom.

That comma is standard written Danish.

I here means in or at the supermarket.

Danish often uses i with places you go inside:

• i supermarkedet
• i banken
• i køkkenet

So i supermarkedet is very natural for at the supermarket.

Supermarkedet is the definite form: the supermarket.

That can mean:

• a specific supermarket already known in the conversation
• or the supermarket as the usual place one shops

Compare:

• i supermarkedet = in/at the supermarket
• i et supermarked = in/at a supermarket

The definite form often sounds more natural when talking about a normal, familiar place.

Because tom is agreeing with flasken, which is singular and common gender.

Predicative adjectives in Danish agree like this:

• common gender singular: flasken er tom
• neuter singular: glasset er tomt
• plural: flaskerne er tomme

So tom is correct because flasken is a singular en-word.

After hvis, Danish uses normal subordinate-clause word order:

• hvis
  • subject + verb
• hvis flasken er tom

So there is no inversion just because the clause begins with hvis.

A useful thing to notice is where adverbs go in subordinate clauses:

• hvis flasken ikke er tom

Here ikke comes before the verb phrase part that follows, which is typical subordinate-clause behavior in Danish.

It is a compound noun:

• opvask = washing up / dishwashing
• middel = agent, substance, product, means

So opvaskemiddel literally means something like dishwashing agent/product, which in normal English is dish soap or dishwashing liquid.

This is a very typical Danish compound: several ideas get combined into one word.