Museet er gratis om onsdagen, og min veninde er heldig, fordi hun har fri den dag.

Museum is the indefinite form: a museum.
Museet is the definite form: the museum.

In Danish, many nouns add the definite ending directly to the noun instead of using a separate word like the.

• et museum = a museum
• museet = the museum

Because the sentence is talking about a specific museum, Danish uses museet.

Om onsdagen means on Wednesdays or every Wednesday.

This is a very common Danish pattern:

• om mandagen = on Mondays
• om tirsdagen = on Tuesdays
• om onsdagen = on Wednesdays

Even though onsdagen looks singular, the whole expression usually has a habitual meaning in English: every Wednesday.

After om, Danish often uses the definite singular form of the weekday when talking about something that happens regularly.

So:

• om onsdagen = on Wednesdays / every Wednesday

This is idiomatic Danish. It does not mean on the Wednesday in normal English. It means a repeated action or situation.

Not in this sentence if you mean a regular schedule.

• om onsdagen = on Wednesdays / every Wednesday
• på onsdag = next Wednesday / this coming Wednesday

So the difference is:

• Museet er gratis om onsdagen = The museum is free on Wednesdays.
• Museet er gratis på onsdag = The museum is free next Wednesday.

Gratis is one of those words that does not change form here. It means free in the sense of costing nothing.

So you say:

• Museet er gratis
• Billetten er gratis
• Billetterne er gratis

The form stays gratis.

Veninde means female friend.
Ven means male friend or sometimes just friend in a more general sense, depending on context.

So:

• min ven = my male friend / my friend
• min veninde = my female friend

Because the sentence says hun later, veninde fits naturally.

Because veninde is a common-gender noun.

In Danish:

• min is used with common-gender nouns
• mit is used with neuter nouns

So:

• min veninde
• min ven
• mit museum would be wrong, because the noun is et museum, but possessives work differently with definiteness and structure

The important point here is simply that veninde takes min.

Because veninde is a singular common-gender noun, and predicative adjectives normally use the basic form with that type of subject.

So:

• min veninde er heldig
• min ven er heldig

But with a neuter singular noun, you often add -t:

• barnet er heldigt

So heldig matches veninde correctly.

At have fri is an idiomatic expression meaning to be off, to have time off, or not to have work/school that day.

So:

• hun har fri den dag = she has the day off that day
• jeg har fri i morgen = I’m off tomorrow

You should learn have fri as a fixed expression. It does not literally mean have free in English.

Yes. Fordi introduces a subordinate clause, and subordinate clauses do not use normal main-clause word order rules in the same way.

In this sentence, the clause is:

• fordi hun har fri den dag

That looks simple, but the difference becomes clearer when you add something like ikke:

• Hun har ikke fri den dag. = main clause
• ..., fordi hun ikke har fri den dag. = subordinate clause

A useful thing to remember is that after fordi, Danish follows subordinate-clause word order.

Den dag means that day or that particular day.

Here, it refers back to the Wednesday already mentioned.

In Danish, when you use a demonstrative like den or det, the noun is often not given the separate definite ending:

• den dag = that day
• det hus = that house

So:

• dagen = the day
• den dag = that day

In this sentence, den dag is correct because it points back to a specific day already mentioned.

The comma before og is there because og joins two full clauses:

• Museet er gratis om onsdagen
• min veninde er heldig

The comma before fordi is connected to Danish comma style. Many writers use a comma before subordinate clauses such as ones introduced by fordi.

So the punctuation is normal Danish, and especially the comma before fordi depends partly on accepted comma conventions.