No, как does not decline. The word here is какой, which declines like a normal adjective. Masculine prepositional is indeed каком.
This site is worth recommending again and again (and I do :-) ) You plug in a word and it will show you all the forms. http://starling.rinet.ru/cgi-bin/morphque.cgi?flags=endnnnnp
Good question. Technically yes, but "state" is almost never used with this meaning in English, usually only in very formal contexts. Generally we'd say "country". You can argue that maybe "state" should be accepted, but all the Americans would think that means New York or Texas, which is not a correct interpretation of страна. I think it's defensible to only accept the common word "country" but if you think "state" should be accepted you can report it.
Который is usually used to refer a single object, which you mentioned just right before: http://www.lingvo-online.ru/ru/Examples/en-ru/%D0%BA%D0%BE%D1%82%D0%BE%D1%80%D1%8B%D0%B9
or to refer one of several objects mentioned before: http://www.lingvo-online.ru/ru/Examples/en-ru/%D0%BE%D0%B4%D0%B8%D0%BD%20%D0%B8%D0%B7%20%D0%BA%D0%BE%D1%82%D0%BE%D1%80%D1%8B%D1%85
It is used also in collocations, such as который час and который год.
You can use который из in questions to select one object from a few of them: "Который из них (этих двух)?", but you can say "Какой из них?" or "Кто из них?" (о людях) as well.
You should not use который in questions about countries, because there are too many of them.