Icelandic Poppies are native of North America rather than Iceland. I am not aware of any exclusively native Iceland plants.

I find this poppy variety self sows very easily and the seed remains viable in the garden for several seasons.
The plants have a long tap root which may make them harder to transplant.
If the root gets broken when trying to dig out the plant a new one will probably grow in its place. Due to this and its profligate seeding I treat them like weeds.

Other Icelandic Poppy Images from Google

The red poppies below are not Icelancid poppies which tend to have finer smooth leaves. They also grow 10-14″ rather than these 24″ flowers.

For an interesting pink poppy see growing garden poppies

Seedheads can be dried to make flower arrangements.

Link for natural Icelandic flora