Garmin Express is so slow and restrictive in the amount of information it allows. Also, it only works when connected to Internet, which is not always available when travelling, therefore many days of records are lost.
No, you can't use Garmin Forerunner 15 with the Training Center because it's not compatible. The Forerunner 15 works by default with Garmin Connect. This option allows you to upload the data to a website to track progress and check various leaderboards. I haven't seen or found anything else about this except Garmin Connect. Additionally, no information could be found regarding compatibility. There is a chance the software isn't compatible with Windows 10 OS.
For additional info, contact support: http://www.garmin.com/us/support/contact?#fragment-1