So, basically, you are trying to make one of these:
:-?
just with an Arduino instead of a 555 timer chip. :-/
Is the one you want to make more elaborate than that? If not, you don't really need more than one servo. The arm on those opens the lid.
Would like to see what you have planned.
Dan
O.K.
Didn't read it all very well. :-[
You can use one servo to open the box, then use a second servo to move the puppet up and swing his arm out. Then you would not have to worry about two servos moving at the exact same time. :-?
Just a thought.