Yes you can use Git on Azure Devops together with other tools. The easiest thing is to install the Git Credential Manager (comes with Git for Windows), it will handle the OAuth workflow.
All of the interaction with Azure DevOps needs to happen from the local -> Clone screen:
I don't know what reasoning the Sourcetree has for not implementing the "Remote" features to list Azure Devops accounts and repositories. The REST APIs are readily available. I suggest you send them a feature request.
But it can work without that. You need to create alternate credentials (git credentials) and set a username+password for your account. You can use those credentials as your username+password for the git repository.
Alternatively, you can create a Personal Access Token with the correct permissions. Then enter any username or simply . or your account name (devops.azure.com/MyAccount => MyAccount) and the Personal Access Token for your password.
Your final option is to use the SSH option in Aure Devops (you need to set it up for your account).
