Correction: #4 is false, this is not the case anymore in Iceland and hasn't been for quite a while. People that immigrate here (to Iceland) can keep their own names and do not need to change anything about them. However, there are rules in place with regards to first and second names of children born here to foreign parents - they are required to have one Icelandic name (first, second or third). Citizens that had to change their names in the past can apply to revert to their original names.
This doesn't mean that you can just name your child whatever you feel like, there are restrictions in place.
Source: http://www.rettarheimild.is/DomsOgKirkjumala/Mannanafnaskra/Um_nofn/#Serstakar_reglur_um_folk_af_erlendum_uppruna (only in Icelandic)
However, there are rules in place with regards to first and second names of children born here to foreign parents - they are required to have one Icelandic name (first, second or third).
Citizens that had to change their names in the past can apply to revert to their original names.
This doesn't mean that you can just name your child whatever you feel like, there are restrictions in place.
Source: http://www.rettarheimild.is/DomsOgKirkjumala/Mannanafnaskra/Um_nofn/#Serstakar_reglur_um_folk_af_erlendum_uppruna (only in Icelandic)