1-)You can create a Virtual Network and after that, create your VM's and set them to use this virtual network. Doing that, your vm's would call the vm with your sql server.
2-) You'll be billed for network, storage and vm compute (including vm size). The best you can do is create a vm and test for a little (with this settings). So you'll know how much your are spending.
3-) When you create a vm, the vhd is stored in a storage account. There's a backup for that, but you can download the vhd and make a version of that on premises.
4-) My suggestion, attach a extra disk to your VM and use it to SQL Server Backup.