1) The database should not exist already. If it already exists, it will notify you from it (could be from a previous installation)
2) Error 25 means that there is a problem, most likely permissions. Are you using Windows authentication for the service account to connect to the remote server? If you are, how exactly did you gave those rights on SQL. I can read it wrong, but it looks like you gave the service account local admin rights on the windows box, and created a SQL login for it through the SQL management studio. If that is the case, that won't work. That service account needs to be added and then given specific permissions.
From our manual:
The account used to run Veeam Backup Service requires db_datareader and db_datawriter roles as well as permissions to execute stored procedures for the configuration database on the Microsoft SQL Server. Alternatively, you can assign db_owner role for this database to the service account.