Conversions in Customer.io track the last email or communication sent before the user enters a conversion segment. If you are tracking users performing an action or entering a segment after receiving a push notification, then a user can be marked as converted after receiving the notification.
You would want to view successfully completed messages in the Campaign Sent tab, which will show a state of Sent or Failed. You can also filter for these push notifications in the Delivery Log.
Yes! To segment for a push notification, you can use the segment builder as usual: