Welcome to Android. I will try and answer your questions the best I know how. Bare with me as I explain.
Apps are basically programs that run on your phone OS. In this case your OS is Android. Widgets are a type of app that allows you to display updates for that app on your home screen. For example: I have the “Weatherbug Elite App.” (Which includes widgets.) I have placed on my home screen the next day forecast and radar maps widget. The widgets update on there own without me doing anything. So I can see at a glance, current weather and local radar. If I want to see more, I just click the widget and it takes me into the Weatherbug app.
Not all apps have a widget option and some apps are widgets only. (You can find this out by just checking the apps description.)
Question 1) In the Android market, I believe all the apps listed, under the widget tab, are either widgets only or apps that have the option to select it as a widget to place on your home screen. (Like my example above.) I don’t see an option to filter it down anymore.