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

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)

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

Can one classify all periodic modules over this algebra?