I am aware that assigning the type of Type to be Type (rather than stratifying to a hierarchy of types) leads to an inconsistency. But does this inconsistency allow the construction of a welltyped term with no normal form, or does it actually allow a proof of False? Are these two questions equivalent?
Girard's paradox constructs a nonnormalizing proof of False. You could read Hurken's "A simplification of Girard's paradox", or maybe Kevin Watkin's formalization in Twelf. In general, these questions are not equivalent, though they often coincide. A "reasonable" type theory will by inspection have no normal proofs of False, and so then normalization implies consistency. The inverse (nonnormalization => proof of False) is much less obvious, and it is certainly possible to construct reasonable paraconsistent type theories, where nontermination is confined under a monad and does not result in a proof of False. 


It leads to a welltyped term, having no normal form, which is assigned the type False. You can find the term given explicitly in A simplification of Girard's paradox 

