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The Norwegian travel document requirements apply to the whole country, not just the mainland, even though border checks take place between the mainland and Svalbard. So yes, a Dutch ID card is sufficient
Upon arrival in Oslo, you will clear mainland Norwegian immigration. There are two different sets of immigration booths for entering and transit passengers. If your tickets are separate and you have checked luggage, clear entry immigration to pick it up and re-check it. Otherwise clear transit immigration (signposted "Transfer to non-Schengen destinations").
If having cleared entry immigration, you will also clear exit immigration to reach the gate.
Svalbard, however, only sporadically performs border checks