Let $(\mathcal{M},E)$ be an internally non-well-founded model of set theory i.e of $ZFC^{\neg f}=ZFC\setminus \mathrm{foundation}+\neg \mathrm{foundation}$, then $\mathcal{M}$ includes an infinite decreasing $E$-sequence. I am interested to know about the degree of illness of internally non-well-founded models in the literature. For example we see there are many models of $ZF$ satisfying various versions of $AC$ that are really distinct. My question is:

$*$) What is the difference between internally non-well-founded models, in the sense of axiom of foundation? I mean how the existence of different decreasing sequences in different models effect their universes. Does an especial sequence capture some interesting properties in that model, in which not satisfying necessarily by all internally non-well-founded models?

I am also interested to find the answer of the following question.

$\bigstar$) Is any of the following statements true?

$(\rm{I})~~~~$ Working in $V$, for any infinite ordinal $\beta$, there exists a model $(\mathcal{M},E)$ of $ZFC^{\neg f}$ with $Ord(\mathcal{M})=\beta$ such that $\mathcal{M}$ contains a decreasing $E$-sequence of length $\beta$.

$(\rm{II})~~~$ Working in $V$, for any infinite ordinal $\beta$, there exists a model $(\mathcal{M},E)$ of $ZFC^{\neg f}$ with $Ord({\mathcal{M}})=\beta$ such that for any $\alpha<\beta$, $\mathcal{M}$ has a decreasing $E$-sequence of length $\alpha$.

clearly $\rm{I}\longrightarrow\rm{II}$.

Edit: I have been thought that the concept of an ill-foundeded model and aninternally ill-foundeded model are the same, but Prof. Enayat and William informed me about the difference, in the following comments. The question $(\bigstar)$ answered by Prof. Enayat stems from the question $*$. I thought maybe $\bigstar$ shows me some different pictures about non-well-founded models.

internally non-well-founded, orinternally ill-foundedas a way of referring to models of $ZFC^{\neg f}$. I also suggest deleting the sentence about compactness, since compactness is of no use in building models of $ZFC^{\neg f}$ , compactness can only be used to build models that are non-well-founded from an external point of view. $\endgroup$ – Ali Enayat Jan 1 '16 at 14:34