Yeah, because there was a convenient public toilet right there and the soldier reached over its wall. Or, maybe, the "vinegar" was the ration of sour wine the soldiers carried.
The point is relevant whether that actual sponge on a stick came from a toilet or not. The contempt implicit in the giving is not at all clear if you don't know what a sponge on a stick was usually for.
Not really? It would be notable if I handed you a slice of pizza sat atop a wad of toilet paper, even if it's just unused paper and not actually from the bathroom.
It's not like Jesus was going anywhere if the soldier didn't come back immediately.
Anyway, the text implies that the sponge was soaked first, and then put on a stick (maybe because the cross was too tall). If it had been an actual xylospongium it would have been already on a stick.