As someone who has worked with Xamarin (certified) for over 2 years, I will try to answer your questions:
1) Xamarin is a .NET platform that compiles native code for both platforms. 2 years ago you created projects for each type of your application (Android, iOS, etc.). In practice, this meant that you write "Actions for android" (in the .NET way to do something, but with the same native Android types) and UIViewControllers for iOS. Xamarin.Forms has not been working for a long time and is a solution that will allow you to share the user interface codec, as well as business logic.
2) Depending on the application, either Xamarin.Forms or its own route are selected. In the case of a native route, you will create views for each platform (Android AXML and iOS Storyboarding). In addition, a PCL containing all the business logic will be added. So you can call
var authResult = AuthService.Login(username,password); var settings = SettingsService.GetUserLocalDbSettings();
with both iOS and Android, and the business logic for logging in is only once. Xamarin.Forms also allows you to share the user interface by creating XAML layouts in your Xamarin.Forms project. They are converted to the right screen in any OS. Note. This only works with simple layouts (head lists, tabs, etc.). Complex layouts take longer in forms until you switch to native.
3) Xamarin is the only environment that allows you to write .NET for these platforms. There are many available plugins available (for example, connection check). Install them in your own projects and in PCL, and you can check if the application has the ability to connect from PCL or not). Hamarin is a hammer, Nuget offers nails.
4) You can share a lot of code (PCL). My honest experience is that most of the time you will build layouts (views) for both platforms. Business logic (logging in, saving items to an SQLite database) requires the least amount of time to build. UI / design, however, requires much more patience. So yes, you can share all the business logic (since the libraries used respect the PCL profile).
5) Protip: Xamarin offers several free university studies since they were part of Microsoft.