I wouldn't take the bowling alley reference too seriously, since the bloke was pissed as a fart at the time. He just as well have said there's a derby race in the ballroom. On the other hand, "bowling alley" could be ship board slang for some real facility like the torpedo bay (if the TWoK configuration is any measure) the Hanger Deck or indeed on of the nacelle catwalks.
I just think it's a little TOO much of a luxury to have that specific a leisure facility. More likely you'll have a rec-deck like the one in TMP with holographic games, maybe even holographic bowling.
I like the idea of swim spas rather than a single pool and a few multi purpose gymnasiums would seam to be appropriate.
Not sure I agree with the idea of a general store though. Technically it's still a military ship (broadly speaking) and with the absence of money I imagine most material requirements would go through the Ship's Quartermaster as would any "snail-mail".
I think in TNG we saw something vaguely resembling a store on the E-D but that was basically a huge replicator for making giant bunnies. For "shopping" purposes I imagine it's usually more a case of the crew buying/trading things on shore leave than the other way around.