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Which two countries share the shortest international land border?

Vatican City and Italy

The Vatican's entire perimeter is its border with Italy, roughly 3.4 km. Monaco's border with France is about 6 km, the next shortest.

Which territory has the highest population density in the world?

Macau

Macau packs roughly 20,000 people per km2 into about 33 km2. Monaco is comparable; among larger sovereign states Bangladesh and Singapore lead.

Which two countries have square national flags?

Switzerland and Vatican City

Both are 1:1. Switzerland's civil ensign at sea is rectangular, because square flags do not behave well on ships.

Which country has the youngest population by median age?

Niger, at around 15

Niger's median age is around 15, meaning half the country is a child. Monaco and Japan are at the other end, above 48.

Nauru holds which distinction?

The smallest republic

Nauru covers about 21 km2 with roughly 12,000 people. Only Monaco and Vatican City are smaller, and neither is a republic.