Auk þess að prútta skoðum við forngripina og varninginn.

Here auk þess að means besides, in addition to, or as well as.

So Auk þess að prútta ... means Besides bargaining / In addition to bargaining ...

A very useful pattern is:

auk þess að + infinitive

which means besides doing something.

For example:

• Auk þess að vinna lærir hún.
  Besides working, she studies.

Because this construction uses að + infinitive.

• prútta = to bargain / to haggle
• að prútta = to bargain

So in auk þess að prútta, the speaker is talking about the action in a general way: besides bargaining.

This is similar to English using a gerund (bargaining) or an infinitive-like structure depending on the expression.

This is because Icelandic normally follows a verb-second pattern in main clauses.

The sentence begins with the fronted phrase:

Auk þess að prútta

Once that phrase takes the first position, the finite verb usually comes next:

Auk þess að prútta | skoðum | við | forngripina og varninginn

So skoðum comes before við.

If the sentence started with the subject instead, you would get the more basic order:

Við skoðum forngripina og varninginn ...

Skoðum is the 1st person plural present tense of skoða.

• skoða = to look at / examine / inspect
• við skoðum = we look at / we examine

So skoðum matches the subject við = we.

They are in the accusative case because they are the direct objects of skoðum.

The verb skoða normally takes an accusative object:

• skoða eitthvað = look at / examine something

So:

• skoðum forngripina
• skoðum varninginn

both use accusative forms.

Those endings show both case and definiteness.

Icelandic usually adds the definite article as a suffix, instead of using a separate word like English the.

So:

• forngripina = the antiques
• varninginn = the merchandise / the wares

More specifically:

• forngripina is a definite accusative plural form
• varninginn is a definite accusative singular form

That is why they do not appear in their basic dictionary forms.

The dictionary forms are:

• forngripur = antique / artifact
• varningur = merchandise / goods / wares

In the sentence, they appear in inflected forms:

• forngripina ← from forngripur
• varninginn ← from varningur

This is very normal in Icelandic, since nouns change form for case, number, and definiteness.

Because the two nouns are being used in different ways.

• forngripina means the antiques — a countable plural
• varninginn means the merchandise / the wares — often treated as a collective noun

So even though English might sometimes use a plural like wares, Icelandic can use singular varningur to refer to goods collectively.

Yes. A version with more neutral word order would be:

Við skoðum forngripina og varninginn auk þess að prútta.

That is also grammatical.

The original sentence puts Auk þess að prútta first for emphasis, as if saying:

Besides bargaining, ...

So the difference is mostly about focus and word order, not basic meaning.

Not always. It can stand on its own, as it does here:

• að prútta = to bargain / to haggle

But you can also add more information, for example:

• prútta um verð = haggle over the price
• prútta við sölumanninn = haggle with the salesman

So in this sentence, prútta is being used in a general sense: bargaining.