I have several questions on the exterior algebra of a vector space:

Q1:When has the exterior algebra A (viewed just as an algebra, not considered as a graded algebra) of an $n$-dimensional vector space over a field $K$ the structure of a Hopf algebra? (depending on n and K)

Q2:Possibly harder question: Is there a finite dimensional, nonprojective module M over A with $Ext_A^{1}(M,M)=0$?

(Q2 is also open in the graded case and has a positive solution in a special case, see the last chapter of https://arxiv.org/pdf/1701.01149.pdf )

Q3:Can one classify all periodic modules over this algebra?