Europe trip for a young family

Well, I won't comment on Disneyland, since I am not a Disney fan.

With kids that young, you may want to think about spending time outside of cities. In 2011, i took my family to Europe. All the kids were older than 20. Even so, we stayed one day in an "agro-tourismo" - a farm with a house to house skiers, but this was summer, so we had it all to ourselves. They had chickens, cows, etc. My kids loved it (20+). They spend a lot of time fooling around with the chickens. For your younger kids, they would probably really enjoy that.

"Any advise on the overall itinerary."

The flights you are taking, once you include transit time to/from the airports and extra time for security check-ins and waiting time, will require much more time than you might think on those travel days - flying doesn't completely "solve" the problem of having destinations that are too distant from each other. Then you have outings that involve substantial ground travel as well. If you must stay outside the city centers, you have additional cash and time drains.

You can look into D'land costs on your own, I imagine. I think you will find accommodations for 4 in Zurich difficult in your price range.

It sounds like Paris and Switzerland are musts. I suggest you keep them but stay out of Zurich except for your final night before flying home; visit Bern and the Bernese Oberland instead.

Instead of Prague and Dresden, visit some of the nice places that lie BETWEEN Paris and Switzerland, places you can reach easily by train. Old-world STRASBOURG is less than two hours from Paris. To see a little of Germany, just catch a train east from Strasbourg across the German border to the Black Forest ("Schwarzwald") region... visit GENGENBACH and some towns along the Black Forest Railway, like GUTACH, a VERY nice place for families. FREIBURG, not far from the Swiss border and on the route to Bern, is also a pleasant city.

You will find lodging relatively inexpensive in the Black Forest; farm stays like Paul mentions and apartments are fairly numerous.

http://www.blackforest-tourism.com/stg/ukv
http://www.bensbauernhof.com/ (See Schwarzwald link.)

Note that if you overnight in a town like Gengenbach or Gutach, your train travel within the region is free of charge, even to the Swiss border, thanks to the KONUS program:
https://www.dreisamtal.de/en/service/konus.php?lang=en

We spent a day at Disneyland Paris in July. Kids are 12 and 9 and we all loved it. We booked our tickets online and that saved us some money and some time when we got there. Have you ever been to Disney in the US? The parks are much more compact than in the US. We were able to do both parks in one day. It was different enough from the US parks to make it fun for us to experience.
We had 6 days in Paris and this is what we did:
- Eiffel Tower (pre-booked tickets)
-Arc de Triomphe ( pre-purchased tickets)
- Catacombs. I think your kids are too little for this one
- cruise on the bateaux mouche
- Notre Dame Cathedral
- Sacre Coeur and Montmartre neighbourhood
- Versailles: The chateau was surprisingly a disappointment for us but we rented bikes and toured the grounds which saved the day :)
- Jardin des tuileries (highly recommend with kids)

We rented an apartment for our stay.... We love having the extra space and the opportunity to make some meals on site. If this is of interest, you may be able to up your lodging budget by the difference in your food budget... just a thought as I think it may be difficult to find hotels that sleep 4 for 120 Euro a night

Citadines, Adagio are apart'hotels with kitchens.
Ibis, Accor, and Mercure are family friendly options if you don't want to cook.
There will be space for 4 people in all these places.
You might qualify for discount prices at Citadines or Adagio, since your trip is next year.