Snow is rather unlikely in all of your named destinations.
The most likely for snow is Zurich, which is still not guarantied.
We have had snow in the Netherlands (so also Amsterdam) for Christmas and the new year. But it is a very small chance of snow.
Same goes for Belgium, Paris and Frankfurt. Rain is much more likely and often comes with strong winds.
Italy is mostly a bit milder, with an even lower chance on snow and more chance on temperatures above the freezing points. But even there no guaranties, weather in Europe is unpredictable.
Whether you would like being a tourist in that time of year depends on what you are used to.
If you are used to winters well under the freezing point, you will find it pleasant, maybe a bit wet but not too cold.
But if you are used to mild winters that never dip under 15 C, with almost no wind, you may find it rather cold and nasty.
And if you are hoping for snow, plan to go up in the mountains in Switzerland or Italy. Around the end of the year you may find it hard to find any fresh snow, although most years will have some new snow in the mountains in November or December.
When coming from an area that does not have cold weather with driving winds, having just a short time, I would skip Amsterdam and the coastal area of Belgium. If you have a long time, say 2 months for your list, plan the visit to those areas when the weather predictions are the best.
In the other areas on your list you are less likely to have as strong winds in strong rain or it has a lot more to do inside. Paris has the Metro, which will allow you to travel out of the weather, which is much easier on your body when it is cold and windy.