Welcome to My Swiss Alps and the forum.
I have some partial answers to get you started.
First of all, without seeing your specific itinerary, it is difficult to answer your questions. And, it seems from casual observation that you may be trying to fit quite a lot into the days that you are there.
That lodging budget is difficult to meet in big cities or main tourist sites. At least, you will be there before the heavy tourist season begins. Sometimes it is best to stay just a bit outside the main cities, on a train line or in a place with frequent bus service. This thread describes one:
This may help, as it list some budget hotels.
www.myswissalps.com/pl antrip/doityourself/sa vemoney
It is hard to judge what will interest children of ages 4 and 6. My most recent experience was with an 8 year old granddaughter. Shew was fascinated by the Lion in Luzern, and also the Bourbaki Panorama nearby.
There are childrens' playgrounds in many towns and cities in Switzerland. There is an especially nice one near the Transport Museum ( das Verkehrshaus) in Luzern, near the lake shore , accessible by bus and, more fun, lake boat. Or, you can walk along the pleasant lake shore to reach it.
If the link does not give you the aerial image, switch to it.
In the main stations in Bern, Luzern, Zürich, and others, there is a very useful tourist information office. I have used the one in Luzern many many times. I am s ure that they would have ideas for children's activities.
Would they like to experience snow?
Vegeterian food is widely available in Switzerland. Many restaurants have "Veggie Special" or "Veggie Hit"
www.myswissalps.com/ab outswitzerland/culture history/food
One of the oldest vegetarian restaurants in Switzerland and in Europe is "Hiltl" in Zürich.
<<"The restaurant holds the Guinness world records for being the oldest continuously open vegetarian restaurant in the World.">> Could be.
In larger cities, the Coop, Migros and other stores have buffets which include vegetarian items.
<<"Kindly suggest us good accommodation options also near stations (Kids get tired soon).">>
If you choose a Swiss Travel Pass for discounted travel, city buses and trams will be free. The tram and bus service is frequent in major cites..every 8 minutes for trams during the day. So, you may not need to stay close to a main station. There is some form of free or cheap fare for children. Arno and Annika are the experts on travel passes. Perhaps they can jump in.