Mine nye færdigheder i at danne passive sætninger bliver anvendt i denne lektion.

Because mine has to agree with the noun being possessed, and færdigheder is plural.

The forms are:

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

So:

• min færdighed = my skill
• mine færdigheder = my skills

Because the adjective is connected to a plural noun and also comes after a possessive.

In Danish, adjectives often take -e in these situations:

• plural nouns: nye færdigheder
• after possessives: min nye bog, mine nye færdigheder

So nye is the correct form here.

Because Danish normally does not use an article together with a possessive.

So you say:

• mine færdigheder = my skills

not something like:

• de mine færdigheder

This works much like English: my skills, not the my skills.

After nouns like færdighed and evne, Danish often uses i at + infinitive to mean skill in doing something.

So:

• færdigheder i at danse = skills in dancing
• færdigheder i at skrive = skills in writing
• færdigheder i at danne passive sætninger = skills in forming passive sentences

Using just at danne would sound incomplete here, and til at danne is less idiomatic in this specific pattern.

Yes, both are possible.

In your sentence, passive is an adjective modifying sætninger:

• passive sætninger = passive sentences

But Danish also likes compound nouns, so:

• passivsætninger

is also very natural, and many learners will meet that form too.

So these are both understandable:

• passive sætninger
• passivsætninger

The compound form may feel a bit more compact and idiomatic.

Bliver anvendt is a passive construction made with:

• blive
  • past participle

So:

• bliver anvendt = is used / is being used

Danish also has another passive form, the -s passive:

• anvendes

Both can often mean roughly the same thing.

Very roughly:

• bliver anvendt can feel a bit more like an action or process
• anvendes can feel a bit more neutral, formal, or general

So both of these are possible:

• Mine nye færdigheder ... bliver anvendt
• Mine nye færdigheder ... anvendes

Because Danish verbs do not change form for singular and plural subjects in the present tense.

So Danish says:

• jeg bliver
• du bliver
• han bliver
• vi bliver
• de bliver

The form stays the same.

That is different from English, where you get is vs. are.

Anvendt is the past participle of anvende, which means to use or to apply.

So:

• anvende = to use/apply
• anvendt = used/applied

In this sentence, it is part of the passive:

• bliver anvendt = is used

Also, anvende is a bit more formal than bruge.

Compare:

• anvende = use/apply, more formal
• bruge = use, more everyday

So a more everyday version might be:

• Mine nye færdigheder ... bliver brugt

There are two points here:

1. Lektion is a common-gender noun, so the demonstrative is denne, not dette.
2. After denne, the noun stays in its basic form, not the definite form.

So:

• denne lektion = this lesson

not:

• dette lektion
• denne lektionen

Compare:

• lektionen = the lesson
• denne lektion = this lesson

The sentence is structured like this:

• Mine nye færdigheder i at danne passive sætninger = subject
• bliver = finite verb
• anvendt = past participle
• i denne lektion = adverbial phrase

So the finite verb bliver is in second position, which matches normal Danish main-clause word order.

A useful Danish rule is V2: the finite verb usually comes second in a main clause.

For example, if you move i denne lektion to the front, you get:

• I denne lektion bliver mine nye færdigheder i at danne passive sætninger anvendt.

The verb bliver still stays in second position.

Yes, it is grammatical and understandable, but it sounds a bit formal and somewhat textbook-like.

A more everyday version might be:

• Mine nye færdigheder i at danne passivsætninger bliver brugt i denne lektion.

Or, if you wanted a more active and natural sentence:

• I denne lektion bruger jeg mine nye færdigheder i at danne passivsætninger.

So the original sentence is fine, but it has a formal, written style.