If everyone died, would Kansas still exist? Or does Kansas have to have someone recognizing it to exist?

The land mass that we call "Kansas" would still exist (unless, of course, the reason that everyone died was that the entire Earth blew up, or something like that). But Kansas the state is a "socially constructed" object (of the sort discussed by John Searle in his book The Construction of Social Reality ) and would thus no longer exist. On the other hand, there might still be books and maps that refer to that land area as "Kansas", so extraterrestrials visiting Earth later on (or Earth animals that evolve to replace humans as intelligent residents of a future Earth) might be able to refer to it that way. Most artifacts are like this: The substance they are made of is human- or mind-independent; the use made of them is not---so, a flat tree stump would still be a flat tree stump after all humans die, but if it had been used as a table by some human, it would no longer be a table.

What does it mean to exist?

I agree with Jonathan Westphal that there's no simple answer to your question as you pose it. One (no doubt overly simpleminded) way to approach an answer to the question is to make a list of things that exist and then see if they have any properties in common. But what would you put on this list? You could think of beginning with a list of all of the kinds of individual things that exist: There are people, there are plants, there are animals, etc. That's going to be a pretty long list, but do these kinds of things really exist? Or is it better to say that only individual things of these kinds exist? So, instead, you should list all the individual people (Adam, Eve, Cain, Abel, ..., you, me), list all the plants (the rose on my desk, the rose on your desk, etc.), list all the animals (my pet cat Bella, your pet dog Fido, etc.). That's going to be an even longer list. But are there such things? Consider any physical object. We know from physics that it's not really a single...

What makes a bottle to be a bottle? The matter that forms it can't represent the actual bottle without the substantial form of a bottle. On the other hand, the substance itself of the bottle has an accidental form. So what exactly happens when the bottle falls down and breaks into small pieces? Is it an accidental change of the form of the substance (Aquino's 'dough' example) or is it a substantial change that leaves behind only the matter of the bottle?

Here's one quick answer to your first question: What makes a bottle a bottle (more precisely: what makes something a bottle) is whether someone uses it as a bottle, not what it's made of or what its form is. Although the rest of your question is stated in Aristotelian and mediaeval terms of substance and accident, I think that part of your question concerns the nature of artifacts. On that topic, you might take a look at my colleague Randall R. Dipert's Artifacts, Art Works, and Agency (Temple University Press, 1993).