I think, given the relatively few long-ish trips are best covered by single tickets or single daypasses. For the ‘local activities’ on all days other than 1, 5 and 12, you might even get a local tourist’s card which may include public transit within a certain region, depending on your exact place where you stay. For trips on days 1, 5 and 12, i recommend you to get Saver’s daypasses. They are quite cheap, but they are not refundable and have flexible pricing. When buying far ahead, this is the cheapest option.
For all three children, you can get a co-travelcard which is valid 1 year. You will need to select one of the adults (not neccesairily a parent). The child can then always travel together with that parent without needing an additional ticket. It is, however, not possible to purchase it online as the identity of the child and the accompaniyng person need to be verified.
Excursion suggestion when staying in Brunnen: Take a boat (there are even some steam-powered ones) to Vitznau and then take the rack railway up to Mount Rigi (The oldest rack railway in Europe!). Both is included in a Saver’s daypass. Furthermore, the Stoosbahn (The steepest funicular in the world) between Schwyz and Stoos SZ is also included in the saver’s daypass.
I recommend to buy the tickets via the official SBB website. Do you have a specific problem where it doesn’t work? I have never had problems with it…
Just on a side note: The train station at the airport is called Zürich Flughafen. The train station of Kloten is quite far from the acutal airport and has much less and worse connections.