It only counts if you are playing stroke play. I made a putt in the putting challenge hole #8 at 260 ft, but my report says my longest putt is 53 ft.
Yes, the play is gone for good. You cannot go back and save old shots.
The good thing about recording only stroke play stats is that players generally will play the easier courses in stroke play and use the more difficult courses for Alternate Shot. That way, if you play poorly in A/S it won't count against your overall stat record. It will only record your Won/Loss record. By doing this I've been able to become familiar will all courses. But if I play stroke play, I will generally just play St Andrews course.
One thing I find frustrating is that some players are at LEVEL 90 and have an AVE score of less than 70.0, but they have only played a minimal amount of stroke play. This can be very deceiving because you expect a player with those stats to be a fairly solid player only to find out they are using starter clubs and couldn't hit the broad side of a barn.