I might give it a bit before inspecting the original hive. A new queen might take a little while to start laying and if you inspect right away you might conclude that you're queenless. Nonetheless, after a week or so I'd do a full inspection. I don't mind pulling apart the bottom deeps and going through them. Just take your time and it'll be fine. As far as the super goes, if you need it elsewhere then go ahead and use it. Even if the colony swarmed, the remaining bees will gather nectar during a flow and if they need the space, they'll use it. If they're not using it now they probably will be when the flow gets going again. I'd be tempted to just let it be. Finally, I'd keep an eye on the swarm you collected. I've had a few over the years that supersede their queen soon after I collect the swarms and I always try to be careful about checking to make sure that the supersedure is successful (pretty much like your checking the original colony for a queen). Keep us informed!