Við lesum flokkunarleiðbeiningarnar í hvert sinn sem við flokkum bylgjupappa.

Because the sentence has two clauses:

• Við lesum flokkunarleiðbeiningarnar = We read the sorting instructions
• sem við flokkum bylgjupappa = when/that we sort corrugated cardboard

Each clause has its own verb, so Icelandic normally states the subject again, just like English does in we read ... every time we sort ....

Lesa is the dictionary form, meaning to read.

Lesum is the present-tense form for we:

• ég les = I read
• þú lest = you read
• við lesum = we read

So Við lesum means we read.

It is one compound noun:

• flokkunar- = of sorting / sorting-
• leiðbeiningar = instructions
• -nar = the (plural definite ending)

So:

• flokkunarleiðbeiningar = sorting instructions
• flokkunarleiðbeiningarnar = the sorting instructions

Long compounds are very common in Icelandic.

In Icelandic, the definite article is usually attached to the noun as a suffix instead of being a separate word.

So:

• leiðbeiningar = instructions
• leiðbeiningarnar = the instructions

The same thing happens inside the full compound:

• flokkunarleiðbeiningar = sorting instructions
• flokkunarleiðbeiningarnar = the sorting instructions

This is one of the big differences from English.

Literally, it is close to in each time that.

In natural English, that becomes:

• every time
• each time
• whenever (depending on context)

So í hvert sinn sem við flokkum bylgjupappa means every time we sort corrugated cardboard.

Because sinn is a neuter singular noun, and hvert agrees with it.

Also, this expression uses the accusative:

• sinn = time, occasion
• hvert sinn = each time / every time

So hvert is the neuter singular form that matches sinn here.

Yes. They come from the same basic idea: sorting/classifying.

• flokka = to sort, classify
• flokkum = we sort
• flokkun = sorting, classification
• flokkunar- = the compound form meaning sorting- or of sorting

So the sentence uses the same root twice:

• flokkunarleiðbeiningarnar = the sorting instructions
• við flokkum = we sort

The dictionary form is bylgjupappi.

In this sentence it is the direct object of flokkum, so it appears in the accusative singular:

• nominative: bylgjupappi
• accusative: bylgjupappa

That is why the ending changes.

It is another compound:

• bylgju- = wave / corrugation-related form
• pappi = cardboard / paperboard

So the literal idea is something like wavy cardboard, which matches corrugated cardboard.

This kind of descriptive compounding is very common in Icelandic vocabulary.

A useful overview is:

• Við — nominative, because it is the subject
• flokkunarleiðbeiningarnar — accusative, because it is the object of lesum
• í hvert sinn — accusative in this fixed expression
• við — nominative again, as the subject of the second clause
• bylgjupappa — accusative, because it is the object of flokkum

So this sentence is a good example of nominative for subjects and accusative for direct objects.

Yes. You could also start with the time clause:

• Í hvert sinn sem við flokkum bylgjupappa lesum við flokkunarleiðbeiningarnar.

That still means the same thing: Every time we sort corrugated cardboard, we read the sorting instructions.

Notice that in the main clause, when something else comes first, the finite verb still comes early:

• ... lesum við ...

That is a common Icelandic word-order pattern.