@AsafKaragila in "kenneth kunen set theory an introduction to independence proofs" he uses $\sf ZF^-$ to $\sf ZF$-regularity, I will change it to "$\sf ZF$-regularity" in the question to make it clearer.

Oh, each atom as an omega sequence of ordinals, I understand now, thanks!

Hi Asaf, thanks for answering, you wrote "the atoms are indexed by countable sequences of ordinals, so that there are always surjections onto every set but there is no omega sequence", how does the first and second part works together? And how does it gives you surjective onto a set like omega2?

Where can I find Sierpinski proof for the kappa pairwise non-isomorphic suborderings?

@JoelDavidHamkins thanks!, about the second part, you used that fact the next line, you can use this fact because $\kappa$ is already well ordered?

Can you explain to me how you got $\kappa^+ \le^* 2^{\kappa^2}$? What do you mean codes there? Also, don't you need choice for $\kappa^2\sim\kappa$?