first arduino project. how to implement my dough monitoring system?

I think your component choice is good. Any basic Arduino should do. I'd go with the UNO because there is no reason to use anything less common.

Note that a distance reading of x/2 (where x is initial distance to dough) doesn't mean your dough has doubled. Container height - x = initial dough height. When distance = container height - ((container height - x)*2) then your dough has doubled.