My husband and I were in Vienna, Salzburg, and Hallstatt this past September. We went up to the Ice Cave and loved it, but you need to factor in the time waiting to go up on the cable car and the time for your tour to start. We got right on the very first cable car of the morning, but we were in the second tour group so we had to wait at the entrance to the ice cave. Plus, there's quite a steep hike from the cable car to the first set of caves. There's also a fun playground and a couple of small museums (before or after the hike) and some other caves higher up that kids might like. We were there for the better part of a day and would have stayed longer but the weather turned cold and foggy so the views were obscured and the path got slippery.
We also toured the salt mine and loved it. Again, you need to factor in the time waiting for the funicular, hiking from the funicular to the mine entrance, waiting for your assigned tour group, and then walking back to the funicular. There's a fun viewing platform that looks down on Hallstatt, and we spent some time there and in the little museums along the hike from the mine. We spent at least 5 hours for the salt mine and its environs.
Hallstatt itself is absolutely gorgeous and we enjoyed just sitting by the lake, watching the swans, and appreciating the scenery. We took a very interesting town walk that Rick mentions in his guidebook. We were the only people on the tour, but it was still a good hour of walking.
For me, this is way too much for one day, although technically it is possible. I live at altitude and I hike a lot in the mountains, but I saw people all around me who needed to stop to rest. If you can, omit one activity or do it all by spending the night in or near Hallstatt.