From a purely geometric point of view, the buttons 
, 
and 
are sufficient to build all the (geometrical) others. From a practical point of view, in order to obtain the points on the line for future use, the button 
is also needed. We have not proved this set of axioms is minimal and does not lead to contradiction. We also do not prove it is the only possible set of axioms.
However, the arrangement of the buttons in the application GeoGebra is quite unconnected with the traditional Euclidean hierarchy of axiom and theorem, and the mutual dependency of the theorems themselves.
I have found it to be an interesting exercise for students to disentangle and arrange for themselves relationships and mutual dependencies. It was surprising to me how much time this activity needed. In particular how challenging students from a variety of ages (ages 13-21) found "apparently simple" constructions. Without performing all these classical geometric constructions for themselves there is nothing on which to form the basis of classification. False starts, the use of buttons "unnecessarily" and the time needed for group discussion expatiate these problems. Justifying why a construction is correct is a further, and arguably a mathematically more challenging, task. The problem is somewhat open-ended and the level of complexity can be varied from simply asking for how a single button works, to a complete classification. The conic constructions are more advanced, and showing that is a theorem, not an axiom, is a significant challenge.