Here is what is known (or at least widely believed) about your two questions.

1. Uniform‐supercompact ⇒ Reinhardt in pure ZF?
   Nobody has been able to prove or refute this in just ZF. All of the familiar arguments that “if
   j∶V→M is elementary and M is closed under Ord‐sequences then in fact M=V and so j is a Reinhardt
   embedding” collapse somewhere to an appeal to a choice‐like principle. For instance one shows that
   M has a global well–order (by enumerating every set by an Ord–indexed sequence), hence AC holds
   in M, and then one restricts j to M and invokes Kunen’s theorem in M. But without any form of choice
   one cannot manufacture in V a definable enumeration of an arbitrary set by ordinals, nor can one
   conclude from j∶V→M elementary that its restriction to M is still elementary for all formulas that
   M thinks true. In fact the consensus is

   “In ZF alone it is an open problem whether a uniform supercompact κ must carry a nontrivial
   j∶V→V (i.e. be Reinhardt).”

   In particular none of the standard proofs of Kunen’s inconsistency go through in the complete absence
   of choice, so one cannot even show uniform supercompact ⇒ “inconsistent,” let alone ⇒ Reinhardt.

2. Consistency strength of “ZF + ∃ a uniform‐supercompact κ.”
   Since every Reinhardt cardinal is trivially uniform supercompact, you get

   Con(ZF + “Reinhardt κ”) ⇒ Con(ZF + “uniform‐supercompact κ”).

   No strictly stronger upper bound is known, nor is any nontrivial lower bound beyond Reinhardt. In
   particular one does not even know whether ZF + “uniform‐supercompact κ” is strictly stronger than
   ZF + “Reinhardt κ,” or whether they are exactly equiconsistent (or indeed whether either is consistent
   at all).

So the state of affairs may be summarized:

• Under any weak choice‐principle (for instance DC on ordinals, or a global well–order of V), one can
show that a uniform supercompact embedding j∶V→M forces M to satisfy ZFC and forces j↾M to be a genuine
self–embedding of M; hence Kunen's proof goes through and one sees uniform supercompact ⇒ Reinhardt ⇒
inconsistent.

• But without any form of choice the step “M is well–ordered” or “j↾M remains elementary for M‐truth”
cannot be carried out, and so both questions—(1) “uniform ⇒ Reinhardt” and (2) “what exactly is the
consistency strength?”—remain open in ZF.

References for further reading:

– Fuchs–Hamkins–Reitz, “Set‐theoretic geology,” where some of the choice‐free issues in embeddings are
discussed.
– Suzuki, Usuba, and others on “Kunen inconsistency without choice” for partial results under DC.
– The survey article by Foreman on large cardinals without AC, which spells out exactly where the devil
lies in the coding arguments.