The name on the baseball shirt that Colonel Kira is wearing changes from "Kira" to "Nerys" throughout the game.
When Sisko dives to catch a bouncing ball a few feet off the ground at 30:30, a longer shot has already established that the ball is rolling or bouncing very close to the ground, not up where Sisko catches it.
Colonel Kira is reading the PADD and reads that if a pop fly is caught it's an out if there are fewer than two outs. The criteria is that the ball has to be caught before it hits the ground so it doesn't necessarily have to be a pop fly it could be a line drive.
When Ezri Dax back flips off the wall to catch a high fly, it is obviously a male stunt double performing the move.
As the Federation anthem is playing, both teams are holding their caps over the upper left chest. This is not where the Vulcan heart is located. However, as Vulcan's would assign no significance to the "emotional symbolism" of the hat placement, it can be inferred they are simply adopting the observed form.
Despite previous episodes showing that Dr Bashir has genetically enhanced hand eye coordination (eg when playing darts with O'Brien), it is not mentioned in this episode, nor is the trait present at all as Bashir struggles to get a hit on the ball. One would expect someone genetically engineered like Bashir to easily hit a home run.
Unless this is another example of the Doctor faking it?
At times the team are standing at impossible distances from each other, the holosuite is a small room in a physical space.
A player missing a base (including home plate) does not need to be tagged with the ball. Instead, the defensive player just needs to tell the umpire "he missed the plate (or whatever base)", and with the ball in hand or glove, step on the missed base to record the out. Additionally, the Vulcan base runner should have been considered out as soon as he left the field while the ball was still in play, but both of these gaffes are probably the result of the players and umpire not fully understanding the rules.
In the post game party at Quark's, a Vulcan player, standing behind and between Quark and Leeta, can be seen with a broad smile laughing at all the fun.
In the Ward Room, Sisko gives the crew a hypothetical baseball situation... Runner on 1st & a runner on 2nd, 1 out, ground ball hit to 2nd base. He then asks Kira what to do. She responds by saying "I go for the double-play, unless the runner on 3rd is already on his way home..." Hopefully by the time the ball reaches 2nd base the runner that was on 2nd wouldn't make it to 3rd. For this instance it doesn't matter if they did or not... Kira screwed up saying there already was a runner on 3rd. The solution in that instance would be to try the double play... touch 2nd then throw the ball to 1st, or touch 2nd then throw to 3rd.
When Quark asks when tryouts are, he has to be told that they're at 13:00 in holosuite 4. As Quark owns the holosuite and is a shrewd and meticulous businessman, he would have known what time Captain Sisko reserved the holosuite and which holosuite it was.