Using windows authentication with Azure WebApp/Website

No, it's not possible. In order to have Windows Authentication is something for on-premise deployments. For Azure Web Sites Azure Active Directory is clearly the best option. Sync from AD to Azure Active Directory is also quite easy to setup.

If you still want to absolutely use Windows Auth and host your website on Azure, you can create Windows VM and host your website there. You then need to join the VM to your AD. To this, both VMs must be in the same network. So if your VM is on-premise you will need to create an site-to-site VPN

Yes, you can connect your web app to Azure Active Directory.

The simplest option is Azure Easy Authentication

http://azure.microsoft.com/en-us/documentation/videos/azure-websites-easy-authentication-and-authorization-with-chris-gillum/.

That's a one click way to authenticate your site.