I believe 2 is equivalent to 3. I'll give a proof for fields.  I suspect this generalizes to algebras but I need to think.  Suppose that $K$ is an infinite extension of $k$.  Then $K\otimes_k K$ is not Noetherian and hence not Artinian semisimple (by Hopkins-Levitzki).  This follows from Theorem 11 of P. Vámos, On the minimal prime ideals of a tensor product of two fields, Mathematical Proceedings of the Cambridge Philosophical Society, 84 (1978), pp. 25-35.