The set $H_\kappa$ of sets hereditarily of cardinality less than $\kappa$ is defined as $H_\kappa=\{x||tc(x)|\lt\kappa\}$. What if we define the set $H=H_{Ord}$ of sets hereditarily of cardinality less than $Ord$; $H$ is the class of sets with some ordinal number as there cardinality. Equivalently, $H$ is the set of hereditarily well-ordered sets.

It is immediately obvious $V=H$ is equivalent to $AC$. My questions are as follows:

1. Is it true that $ZF\vdash H\vDash ZF$? Is it true that $ZF\vdash H\vDash ZFC$?

2. More generally, does $ZF\vdash H_\kappa\vDash AC$?

3. What is the consistency strength of the existence of some $j: H\prec H$? Is the existence of such an embedding first order expressible?

(I am obviously working in the context of $ZF$ without choice. I define $|tc(x)|\lt\kappa\leftrightarrow\exists\lambda\lt\kappa(|tc(x)|=\lambda)$)

The set $H_\kappa$ of sets hereditarily of cardinality less than $\kappa$ is defined as $H_\kappa=\{x||tc(x)|\lt\kappa\}$. What if we define the set $H=H_{Ord}$ of sets hereditarily of cardinality less than $Ord$; $H$ is the class of sets with some ordinal number as there cardinality. Equivalently, $H$ is the class of hereditarily well-ordered sets.

It is immediately obvious $V=H$ is equivalent to $AC$. My questions are as follows:

1. Is it true that $ZF\vdash H\vDash ZF$? Is it true that $ZF\vdash H\vDash ZFC$?

2. More generally, does $ZF\vdash H_\kappa\vDash AC$?

3. What is the consistency strength of the existence of some $j: H\prec H$? Is the existence of such an embedding first order expressible?

(I am obviously working in the context of $ZF$ without choice. I define $|tc(x)|\lt\kappa\leftrightarrow\exists\lambda\lt\kappa(|tc(x)|=\lambda)$)