We unfortunately do not have automated notifications for when your music goes live in the stores. But, you will get a link to your album in iTunes as soon as you see, "copy your iTunes link" appear on your album page in TuneCore. Sometimes, the link will not send you to your release in iTunes. This means your release is being reviewed by iTunes. Learn more .
Regarding the "live" and "sent" statuses of your album in your TuneCore account, it can take up to one week from when your content goes live in iTunes and when your content is marked as "live" in TuneCore.
You can search for your music in other stores by going to their website and searching your artist name or release title.