Problems with Advanced Search

I have an advanced search module in the main area of my home page. (Not as a side widget)

There are few things that I cannot figure out so far
1. If I want to search by Category only, I got message “We have found 0 items matching your search.”
2. If want to search by Country only I got the same message
3. The search process is possible only when I choose at least Category + Country + State How can I make searches by Category or by Country alone?
4.When I’m trying to search in the US, I got full list of US States as a drop-down menu, but all states are in some very strange order. How can I make all states to appear in this drop-down down menu in alphabetical order?
5. How can I add “Latest posts” widget to my Home page? I don’t need it the side panel.