First of all, congratulations on giving up TV. It must have taken a lot of self control both by the children and the parents.
The issue regarding puzzles is that many Poskim consider snapping together pieces of a picture to be equivalent to drawing them and assembling a puzzle to be similar to writing. The arguments are too complex for this venue, but there are a significant minority of Poskim who allow assembling a puzzle that is meant to be taken apart at its conclusion and not framed etc. So, while it is certainly preferable for adults to refrain from assembling puzzles on Shabbos, one should not prevent children from playing with puzzles. Under circumstances such as yours, there may even be grounds to allow adults to participate.
I don't see any problem with the game "Don't break the Ice", but maybe we are thinking of different games. The Hasbro game where you tap out squares of plastic that fit tightly together is completely permitted on Shabbos.
All this only applies to Ashkenazim, however for Sefardim the whole issue of games on Shabbos is much stricter.