I fixed all of these.
The one under debate, "There are moments in a player's career that they will not soon forget" I left alone. It's worded kind of strange which is probably where the confusion is coming from. Putting it into something that one might say normally... "There's moments in a player's career that they won't soon forget" makes sense. Of course "There's" is grammatically incorrect, it should be "there are" or "there're" (which I don't think is a word, probably why it's changed to "there's"). Without the contractions it drags it out and makes it look and sound odd... but I don't want to change it to my line because: 1) I don't want you to post that "There's" isn't right, and 2) it does look like the kind of thing one might read in the paper (I stumble over the writer's trying to make his word quota in the papers all the time). It's also not really talking about only this player, but all players...
I think "back side" would be right. I don't think backside is right though. I changed that. Seems that "backside" would refer to the buttocks, "back side" would be simply the "back side" of his career... as opposed to the "front side".
Not sure about the shutout problem. A no hitter IS a shutout, so it is correct. And it doesn't say it's just a shutout... but that it was a shut out. Actually, reading your clip that doesn't work... it can't be a 4-1 shut out. Fixed that one too.
You can send them to me or post them here, whichever you want. If you send them directly there isn't fun debate over pluralism though.