"Turkey's map" could also imply a map produced by and used by the Turkish government. For example, if you're describing a border dispute between Turkey and another nation, you might say "Turkey's map shows __" meaning that the map that the Turkish government considers correct shows X.
This isn't merely hypothetical. Turkey doesn't have any border disputes as such, but it recognizes a government in northern Cyprus that nobody else does, so Turkey's map shows an independent Turkish Republic of Northern Cyprus, while everybody else's map shows a united Republic of Cyprus.
'Türkiye haritası' is not a full sentence, it is just a noun. The correct translation of your answer is ' O bir Türkiye haritasıdır.' And of course you can also say ' O bir Türkiye haritası'. But in this case, it is like the given word is 'apple' and the translation should be 'elma' not ' O bir elmadır' or ' O bir elma'.
No, türk does not necessarily mean a person, e.g. türk kahvesi, türk bayrağı, türk mutfağı, türk lirasi etc. have no exact meaning of a person in these nouns. Rather something non-human is specified as peculiar to, or characteristic of, a specified nation. Türk is a word expressing Turkish nationality.