På biblioteket viser jeg lånekortet mitt og finner en novelle av en norsk forfatter.

This is because Norwegian follows the V2 rule in main clauses: the finite verb usually comes in the second position.

So when På biblioteket is moved to the front, the verb viser must come next, and the subject jeg comes after it:

På biblioteket viser jeg lånekortet mitt ...

If you start with the subject instead, you get normal subject-first order:

Jeg viser lånekortet mitt på biblioteket ...

Both are correct, but the first version puts extra focus on På biblioteket.

In Norwegian, på is often used for being at a place, especially places you go to for a function or purpose, such as på skolen, på jobb, på universitetet, and often på biblioteket.

So på biblioteket usually means at the library.

By contrast, i biblioteket means more literally inside the library. It focuses more on physical location inside the building or room.

So the difference is often:

på biblioteket = at the library
i biblioteket = in the library / inside the library

The ending -et marks the definite singular of many neuter nouns in Norwegian.

For example:

et bibliotek = a library
biblioteket = the library

et lånekort = a library card / borrowing card
lånekortet = the library card

So in your sentence, biblioteket means the library, and lånekortet means the library card.

Both are possible, but lånekortet mitt is the most common everyday pattern in Norwegian.

When the possessive comes after the noun, the noun usually takes the definite form:

lånekortet mitt = my library card

You can also say:

mitt lånekort

That is correct too, but it often sounds a bit more formal, emphatic, or contrastive, depending on context.

So the most natural neutral choice is usually:

noun in definite form + possessive

like boka mi, huset mitt, vennen min, lånekortet mitt.

The possessive has to agree with the gender of the noun.

lånekort is a neuter noun, so you use mitt:

et lånekort
lånekortet mitt

Compare:

en bil → bilen min
et hus → huset mitt

So mitt tells you that lånekort is neuter.

They are both in the present tense.

The infinitives are:

å vise = to show
å finne = to find

Present tense:

viser = show / am showing
finner = find / am finding

In Norwegian, the present tense can often cover both simple present and present continuous meanings that English separates. So depending on context, viser can mean either show or am showing.

Because both novelle and forfatter take the article en in this kind of Bokmål usage.

en novelle = a short story
en forfatter = an author

A useful note: novelle is historically a feminine noun, so some speakers may also use feminine forms such as ei novelle. But in Bokmål, using en with many feminine nouns is very common, so en novelle is completely normal.

This is because the noun phrase is indefinite singular:

en norsk forfatter

In indefinite singular with an en-word, the adjective usually has its basic form:

en norsk forfatter
en stor bil

You get norske in other situations, especially plural or definite forms:

norske forfattere = Norwegian authors
den norske forfatteren = the Norwegian author

So here norsk is correct because it modifies en forfatter.

In Norwegian, av is commonly used to express authorship or creation, where English often uses by.

So:

en novelle av en norsk forfatter
means
a short story by a Norwegian author

This is a very common pattern:

en bok av Ibsen = a book by Ibsen
et maleri av Munch = a painting by Munch

So here av does not mean of in the usual English sense. It specifically shows who wrote the story.

Because both verbs have the same subject: jeg.

So Norwegian, like English, often avoids repeating the subject when two verbs are joined by og:

På biblioteket viser jeg lånekortet mitt og finner en novelle ...

This means:

I show my library card and find a short story ...

You could repeat the subject, but normally you would not unless you wanted special emphasis or clearer structure. The shorter version is the natural one here.