If you use "-s" it means there are a number of options. If you use the singular noun, it means one thing (one destination). It's really that simple.

It's difficult to answer these kinds of questions because there is no context to them, and they are otherwise very generic and could be written so many different ways. Generally, saying "countries" indicates that it is uncertain where you are going, how long you will be gone, how much money you have available. It's very general. To specifically say "country" would indicate that you have narrowed down your itinerary to only one destination, and now it's a matter of which one it will be. But all of this, again, is without context. The second pairing of sentences doesn't sound quite right either way you say it, unless you were using the words for effect. I may say, "There's something I want to know," and you may answer (for reasons unknown without context), "Oh, what THING is that," stressing the word "thing." But you could also do this with "there are some things" and "what THINGS are those." Context, context, context. If I go to a travel agent and say, "I want to visit Europe," he may ask me, "Which countries would you like to visit?" If I respond, "Well, I'm on a budget (I don't have a lot of money to spend)," he may ask, "Did you have a particular country in mind?" For your second examples, "thing/things" would almost never come into the sentence. We would simply say, "What do you want to know?" The only time I can see specifically stating "things" would be if I were guarded or cautious about the questions that might be asked. For example, I may say to my father, "I have a few questions about our family history," and he may respond, "What do want to know?" He has nothing to hide and will answer whatever questions I ask. However, if a reporter says to Brad Pitt, "I have a few questions about Angelina Jolie," Brad might ask, cautiously, "What kinds of things do you want to know?" This would signify his hesitation to speak too freely and candidly.

"s" is plural. It means more than one. Too simple!

I would correct your questions to "Which country (countries) will you want to go to?" and "What do you want to know?" or "What kinds of things did you want to know?"