I'm going to challenge your basic thinking. there are 4 great cities in EU-- Vienna, London (for now), Prague, and Rome. I've spent 11 days in Prague, 15 days in Vienna, 8 days in London, and 10 days in Rome & I feel like I'm just getting started in each of them (also 6 days in Budapest).
In Prague (my top pick), you need at least 1/2 day for Prague Castle, 4 hours to walk the Royal way, most of day for Old Town, half a day for the Jewish Quarter, 1/2 day for Mala Strana, and then it's time for serious sightseeing & museums. You plan 2 night in Prague, but that includes the better part of day one to get there, leaving only one day a bit to see Prague. Don't do it.
You have been to Vienna. What about the rest of the family? How long were you there (if less than two weeks, it wasn't long enough)?
I don't think you can get from Prague to Budapest via Cesky Krumlov in one day
Budapest is not in the same league as Prague & Vienna,so I strongly advise dropping Budapest (your plan smells of the "If It's Tuesday, it Must Be Belgium" syndrome.
Road trip to Hallstatt: Follow the river there the scenic Wachau Valley to Maria Taferl church, the second best work of the great Baroque archetect, Jakob Prandtaure. Then back track 6 miles to the Abbey at Melk, Prandtaure's masterpiece and in my book, the top sight in Europe. On to the Abbey Church just east of Linz where Bruckner is buried. Kids will be bummed out by the hundreds of skeletons stacked up in the church crypt. From Linz go north thru the Vlatavatal, (Moldau Velley) and lessen in classical music-- The Moldau is the second movement in Smetana's Ma Valast to Cesky Krumlov. Return to Linz. Go west along the river to the great Rococo church at Wilherring, then thru the Salzkammergut to Bad Ischl (Emperor's hunting lodge). Go up the mountain to Bad Ausee. Mahler did most of his composing in small shed (still there) on the shore of the first of three secnic lakes that run east of Bad Ausee. Continue south to the right turn for the road than runs along the north side of the mighty Dachstein. take the cable car up to the glacier and ice caves with great views of the Hallstattersee.
By approaching Hallstatt from th south, you can drive into town. The more common northern approach shunts you into a giant parking garage inside the mountain above the town.
From Hallstatt, go south to the base of the mighty Dachstein, turn right & right again to go north on the road parallel to thewest side of the Hallstatersee, then left and left to the Gosausee. Once you see the Gosausee, you will forget about Hallstatt, a third rate scene in comparison. The Gosausee is the most spectacular Alpine lake in the universe, and nobody knows about it except the natives. There is a hotel on the shore and the lake view rooms are well worth the extra charge. The amazing scenic background for the Gosausee is the west side of the mighty Dachstein massif (the south side is the most scenic).
Although I haven't finished the tour, it's time for bed. If you want the rest, do a reply so indicating.