Mina anteckningar är korta, men hennes anteckning är mycket tydligare.

Because anteckningar is plural.

In Swedish, possessive words like my change depending on the number and gender of the noun:

• min = my, for a singular common gender noun
• mitt = my, for a singular neuter noun
• mina = my, for plural nouns

So:

• min anteckning = my note
• mitt brev = my letter
• mina anteckningar = my notes

Here, anteckningar is plural, so mina is required.

Because the first noun is plural and the second is singular:

• anteckning = note
• anteckningar = notes

The sentence is comparing:

• my notes
  with
• her note

So the grammar reflects that difference directly.

Also, these are the indefinite forms:

• en anteckning = a note
• anteckningen = the note
• anteckningar = notes
• anteckningarna = the notes

Because korta agrees with the plural noun anteckningar.

The basic adjective is kort = short.
Its forms are:

• kort = used with a singular neuter noun, and also as the dictionary form
• korta = used with plural nouns, and also with definite forms

Examples:

• ett kort brev = a short letter
• korta anteckningar = short notes

In your sentence, the adjective comes after är, but Swedish still shows agreement:

• Mina anteckningar är korta = My notes are short

So even as a predicate adjective, it must match the plural subject.

The ending -are makes the adjective comparative.

• tydlig = clear
• tydligare = clearer
• tydligast = clearest

So:

• hennes anteckning är tydlig = her note is clear
• hennes anteckning är tydligare = her note is clearer

In this sentence, it means her note is clearer than your notes, even though Swedish does not repeat än here because the comparison is understood from the context.

Yes. Tydligare already means clearer.

Adding mycket makes it stronger:

• tydligare = clearer
• mycket tydligare = much clearer

So mycket works like much in English when used with a comparative.

Examples:

• hon är äldre = she is older
• hon är mycket äldre = she is much older

Good question. Some Swedish possessives change form, but hennes does not.

These do change:

• min / mitt / mina = my
• din / ditt / dina = your
• vår / vårt / våra = our

But these are usually invariable:

• hans = his
• hennes = her
• dess = its
• deras = their

So you say:

• hennes anteckning = her note
• hennes anteckningar = her notes

The form hennes stays the same whether the noun is singular or plural.

Because in Swedish, a possessive normally replaces the article.

So you say:

• min bok = my book
• hennes anteckning = her note
• våra vänner = our friends

Not:

• en min bok
• den hennes anteckning

This is similar to English: we say my book, not the my book.

Yes. Anteckning usually means a note in the sense of something written down, such as a class note, reminder, or written observation.

Common uses:

• ta anteckningar = take notes
• göra anteckningar = make notes
• mina anteckningar = my notes

So in this sentence, it most naturally refers to written notes.

No, men does not force inversion here.

Men means but and joins two main clauses. After it, Swedish usually starts a new normal main clause:

• Mina anteckningar är korta
• men hennes anteckning är mycket tydligare

So the second clause keeps ordinary main-clause order:

• subject: hennes anteckning
• verb: är

That is why it is hennes anteckning är, not är hennes anteckning.

Yes, that can be perfectly natural.

The sentence is saying something like:

• my notes are short
• but her note is much clearer

So the speaker may be comparing their own set of notes with one specific note belonging to someone else. Grammatically, that is completely fine.

If both were plural, you could say:

• Mina anteckningar är korta, men hennes anteckningar är mycket tydligare.

If both were singular:

• Min anteckning är kort, men hennes anteckning är mycket tydligare.

Yes. The form is the same.

Swedish adjectives can appear:

• before a noun: korta anteckningar = short notes
• after a verb like är: anteckningarna är korta = the notes are short

In both cases, the adjective still agrees with the noun. Since the noun is plural, the adjective form is korta.

So:

• mina korta anteckningar = my short notes
• mina anteckningar är korta = my notes are short