Must all four hydrogen atoms lie in the same plane? If not, what is the spatial relationship among them? Why?

We are considering an Allene molecule here, CH2CCH2. To answer your question, NO, they don't have to lie on the same plane. The spatial arrangement between them is that the center carbon that forms these pi bind in the left and right are PERPENDICULAR to each other.

Explanation:

We see here that The terminal carbons are sp2 hybridized, and form three σ-bonds each which means that each terminal carbon has one unhybridized p-orbital. The central carbon atom is sp hybridized, and forms two σ-bonds which means it has two unhybridized p-orbitals. For better understanding, let's call these two orbitals px and py. Summarily, These orbitals are perpendicular to each other