I think the standard term here, insofar as there is one, for your repeated stacking operation is “Steinhaus [tetrahedron] chain”. Related to your conjecture 1, there is a proof of the fact that no tetrahedron in a nontrivial chain can ever be a translate of the original tetrahedron, in Tomkowicz & Wagon, The Banach-Tarski Paradox (2d ed. 2016), theorem 6.10.
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