Hvis det bliver ved med at regne, bliver vi hjemme og ser en film på internettet.

No. It is the same verb form, but it has two different functions here.

• det bliver ved med at regne = it keeps raining / it continues to rain
• bliver vi hjemme = we stay home

So in the first part, bliver ved med at is a fixed expression meaning continue to or keep on.
In the second part, blive hjemme means stay at home.

This is very common in Danish: the same verb can have different meanings depending on the expression it appears in.

Bliver ved med at means keep doing something or continue to do something.

Structure:

• blive ved med at + infinitive

Examples:

• Han bliver ved med at tale = He keeps talking
• Det bliver ved med at sne = It keeps snowing

So:

• det bliver ved med at regne = it keeps raining

This is a very useful everyday Danish expression.

In Danish, just like in English, weather verbs often use a dummy subject.

• Det regner = It is raining
• Det sner = It is snowing

The det does not refer to a specific thing. It is just required grammatically, like English it in it is raining.

Because after at, Danish normally uses the infinitive form of the verb.

So:

• at regne = to rain
• at se = to watch / to see

In bliver ved med at regne, the pattern is:

• bliver ved med at + infinitive

That is why it is regne, not regner.

This is because Danish uses verb-second word order in main clauses.

When the sentence begins with an if-clause:

• Hvis det bliver ved med at regne, ...

the next part is the main clause, and the finite verb comes before the subject:

• bliver vi hjemme

So the pattern is:

• Hvis ..., verb + subject ...

Compare:

• Vi bliver hjemme = We are staying home / We stay home
• Hvis det regner, bliver vi hjemme = If it rains, we stay home

This inversion is very important in Danish word order.

Because Danish often uses the present tense to talk about the future, especially when the future meaning is clear from the context.

So:

• Hvis det bliver ved med at regne, bliver vi hjemme
  can mean
• If it keeps raining, we will stay home

English often uses will, but Danish very often just uses the present tense in this kind of sentence.

Because hjemme means at home, while hjem usually means home as a direction.

Compare:

• Vi er hjemme = We are at home
• Vi bliver hjemme = We stay at home
• Vi går hjem = We go home

So in this sentence, the idea is location, not movement. That is why hjemme is correct.

Because ser is coordinated with bliver.

The main clause is:

• bliver vi hjemme og ser en film på internettet

Both verbs belong to the same subject, vi:

• vi bliver hjemme
• vi ser en film

In Danish, when two finite verbs are joined with og in this kind of structure, the second one also stays finite:

• Vi bliver hjemme og ser en film

Not:

• Vi bliver hjemme og at se en film

Yes, it is understandable and grammatical. It means something like watch a film on the internet.

That said, in everyday speech many Danes would more often say:

• på nettet = on the net / online

So:

• ser en film på nettet
  may sound a little more natural in casual conversation.

But på internettet is still perfectly correct.

Because the sentence begins with a subordinate clause:

• Hvis det bliver ved med at regne

and then moves into the main clause:

• bliver vi hjemme og ser en film på internettet

In Danish, it is very common to separate these parts with a comma. So the comma helps show the structure of the sentence clearly.

Usually here it is best understood as if.

• Hvis det bliver ved med at regne = If it keeps raining

Hvis introduces a condition. It tells us that staying home depends on the rain continuing.

Danish når is more often used for when in the sense of something expected to happen:

• Når jeg kommer hjem, spiser jeg = When I get home, I eat

So in this sentence, hvis is the right choice because it is conditional.

Yes. Blive hjemme is a standard and very common Danish expression.

Examples:

• Jeg bliver hjemme i dag = I’m staying home today
• Vi bliver hjemme i aften = We’re staying home tonight

So even though blive often means become, in this expression it means stay.