Being an amateur historian, I mostly used Wikipedia info-boxes for dates, though there is some subjectivity to adjudging the continuity of state lineages. I took into consideration name, territory, language and identity. For the phenomenon of "personal unions" and "commonwealths" it is assumed that in practice this meant one entity dominating and continuing their legacy, while the other's was made redundant. Dates of "independence" were calculated at point of first recognition by a significant power, excluding war-time events. I considered Kingdom of Judah to be ancestral to the State of Israel.
Great quiz! I initially thought that Armenia and Georgia wouldn't have gained their independence until 1991, but it looks like both of them became independent briefly in 1918 (Georgia's independence lasted until 1921, while Armenia's ended in 1920, before they were absorbed into the Soviet Union). That being said, Georgia is listed here as gaining their independence in 1922, but it should be 1918 instead (by 1922 they were part of the Soviet Union).
Nice idea for a quiz, even though it's tricky to ascertain continuity sometimes. I'm Egyptian but I was surprised at it coming at number 2. I never thought of how long an ethnic Egyptian state ceased to exit. I think the whole Fatimid/Ayyubid/Mamluk period could be considered an independent Egypt, just not ethnically ruled by Egyptians. Did you only count ethnic independence?
Technically the United Kingdom is short for the United Kingdom of Great Britain and Northern Ireland, which is the updated version of the United Kingdom of Great Britain and Ireland, and before the Kingdom of Great Britain and Ireland united there was technically a personal union between the kingdom of Great Britain and the kingdom of Ireland the same there was a personal union between the kingdom of Great Britain and the kingdom of Hanover. Although by that logic, Croatia should not be on this list because there was a personal union between Croatia and Hungary since 1102 was simply the title "King of Croatia" moving to Hungary's Arpad dynasty.
At the risk of appearing a complete plank, I didn't read the description properly and initially typed in "Canary Islands" and Iceland popped up... presumably it's so people can type in the Icelandic for Iceland? A bit of a gift though (which possibly twerps like me need)