Divine Heir is default, Inner ishin is getting the ending where you die but the heir Lives and you need to get the shura ending to get the shura gauntlet

i got inner isshin with the sculptor ending though.

Dibine Heir is unlocked when you start new game +, shura if you had the shura ending and the other one of your ending was not the shura one. So all 3

I just finished NG+, completing the Shura ending and the ending that requires the Everblossom branch. All three gaunlets are available to me. I think you have to beat all the bosses listed under each gaunlet challenges with the exception of the INNER (Genichiro, owl father & Isshin).

No you don't
Can't farm xp there

Git gut. Well, actually use Malcontent 3x in last phase and divine confetti each phase and it becomes rather easy. Only really difficult one is Inner Isshin (imo)

havent done shura gauntlet but inner isshin was definetly easier than base demon of hatred

you must kill ishin, demon of hatred and owl father in your playthrough to unlock shura