I live in Sweden and I have a national ID card. Can I use it when traveling to Iceland? In security checks, gate checks etc? Or, do I need to show my passport everytime?
Yes, Iceland is closely associated with the EU and all EU citizens can go there with their ID card (if they have one, of course). It's also part of the Schengen area, which means there is no systematic border checks and it's even possible you won't have to show your ID at all.
Depending on airline policy and on the airport, you might still have to show an ID at some stage but your national ID card will be enough.