Regarding Pach's original question: Bing's house deals with the analogous question of collapsibility. It is an example of a contractible non-collapsible space. For that problem it is known that if you allow both collapses and anti-collapses, every contractable space can be reduced to a point. This follows from "Simple homotopy theory". I dont know if this result of simple homolopy theory extends to animals built from cubes but they might.

The modified higher genus question seems easier since you allow steps that changes the topology of the animal.

Regarding Pach's original question: Bing's house deals with the analogous question of collapsibility. It is an example of a contractible non-collapsible space. For that problem it is known that if you allow both collapses and anti-collapses, every contractable space can be reduced to a point. This follows from "Simple homotopy theory". I dont know if this result of simple homolopy theory extends to animals built from cubes but they might.

The modified higher genus question seems easier since you allow steps that changes the topology of the animal.

One can ask a stronger question if when you fixed the genus g you can pass between two animals of genus g by such steps. The analogous question for collapsibility seems to relate to simple homotopy invariants of surface groups but I dont know what they are.

Bing's house deals with the analogous question of collapsibility. It is an example of a contractible non-collapsible space. For that problem it is known that if you allow both collapses and anti-collapses, every contractable space can be reduced to a point. This follows from "Simple homotopy theory". I dont know if this result of simple homolopy theory extends to animals built from cubes but they might.

Regarding Pach's original question: Bing's house deals with the analogous question of collapsibility. It is an example of a contractible non-collapsible space. For that problem it is known that if you allow both collapses and anti-collapses, every contractable space can be reduced to a point. This follows from "Simple homotopy theory". I dont know if this result of simple homolopy theory extends to animals built from cubes but they might.

The modified higher genus question seems easier since you allow steps that changes the topology of the animal.