Let $X$ be a topological space (feel free to add some separation axioms like “completely regular” if they help in answering the questions). Let $\alpha$ be an ordinal, identified as usual with $\{\beta \in \mathit{Ord} : \beta<\alpha\}$. Consider the following sheaves of $\alpha$-valued functions on $X$:
$\mathscr{C}_\alpha$ takes an open set $U\subseteq X$ to the set of continuous functions $U\to\alpha$ where $\alpha$ is given the usual (=order) topology.
$\mathscr{S}_\alpha$ takes an open set $U\subseteq X$ to the set of upper semicontinuous functions $U\to\alpha$, where “upper semicontinuous” means that $f(x) = \operatorname{lim\,sup}_{y\to x} f(y)$ for all $x\in U$ and of course $\operatorname{lim\,sup}_{y\to x} f(y) := \inf_{V\ni x} \sup_{y\in V} f(y)$ where the $\inf$ ranges over all (open, say) neighborhoods $V$ of $x$ in $U$.
$\mathscr{N}_\alpha$ takes an open set $U\subseteq X$ to the set of normal upper semicontinuous functions $U\to\alpha$, where “normal upper semicontinuous” means that $f(x) = \operatorname{lim\,sup}_{y\to x} \operatorname{lim\,inf}_{z\to y} f(z)$ for all $x\in U$ (of course, $\operatorname{lim\,inf}$ is defined dually to $\operatorname{lim\,sup}$).
The restriction maps are pointwise restriction in each case. Since each of these is defined by a local property on functions, they are, indeed, sheaves of sets, i.e., objects of the topos $\operatorname{Sh}(X)$ of sheaves of sets on $X$.
We have $\mathscr{C}_\alpha \subseteq \mathscr{N}_\alpha \subseteq \mathscr{S}_\alpha$ where $\subseteq$ means “is a subsheaf of (for the obvious inclusion maps)”.
Examples:
$\mathscr{C}_\omega$, the set of locally constant $\mathbb{N}$-valued functions on $X$, is the natural numbers object of the topos $\operatorname{Sh}(X)$. And $\mathscr{C}_n$, for finite $n$, is the $n$-fold coproduct $1+\cdots+1$ of the terminal sheaf.
$\mathscr{S}_2$ takes an open set $U \subseteq X$ to the set of open sets $V \subseteq U$ (identifying an open set $V$ with the upper semicontinuous function $\mathbf{1}_{U\setminus V}$) with restriction defined by intersection, i.e., it is the subobject classifier $\Omega$ of $\operatorname{Sh}(X)$. More generally, $\mathscr{S}_n(U)$ is the set of all chains $U \supseteq V_0\supseteq \cdots\supseteq V_{n-1} = \varnothing$ of open sets.
$\mathscr{N}_2$ (as a subsheaf of $\mathscr{S}_2$) consists of regular open sets $V \subseteq U$, and can be identified with the internal subset $\{p\in\Omega : \neg\neg p = p\}$ of $\Omega$.
Each of these sheaves comes equipped with a strict ordering $<$ defined by $f<g$ whenever $f(x)<g(x)$ for all $x\in U$. (There is also a lax ordering $\leq$ defined analogously, and the two satisfy axioms (1)–(6) from this question, and in the case of $\mathscr{C}_\alpha$, even (7). But I believe $<$ is more interesting than $\leq$.)
Furthermore, note that $<$ is well-founded in the sense that if $\mathscr{P}$ is a subsheaf of $\mathscr{F}_\alpha$ (any one of $\mathscr{C}_\alpha, \mathscr{S}_\alpha, \mathscr{N}_\alpha$) such that “any $f \in \mathscr{F}_\alpha(U)$ for which ‘every $g\in\mathscr{F}_\alpha(V)$ with $V\subseteq U$ such that $g<f$ on $V$ belongs to $\mathscr{P}(V)$’ itself belongs to $\mathscr{P}(U)$” then in fact $\mathscr{P}=\mathscr{F}_\alpha$. (This is proved by induction on $\alpha$, and, of course, the case of $\mathscr{S}_\alpha$ suffices. Note that if we had used lower semicontinuous functions, I think this would not work, e.g., among the lower semicontinuous $\mathbb{N}$-valued functions, the locally bounded ones form a subsheaf which appears to satisfy the induction hypothesis but not generally its conclusion.)
Questions:
Can we provide an internal description of $\mathscr{C}_\alpha, \mathscr{S}_\alpha, \mathscr{N}_\alpha$ (i.e., couched in the language of intuitionistic set theory) that makes sense in an arbitrary elementary topos? (Not demanding, of course, that these objects exist in an arbitrary topos: e.g., $\mathscr{C}_\omega$ should be the natural numbers object, which doesn't always exist. Also, not demanding that the description be “uniform” in $\alpha$: it can iterate “externally” over $\alpha$.)
How do these objects relate to the various kinds of intuitionistic ordinals considered in Paul Taylor's paper “Intuitionistic Sets and Ordinals” (J. Symbolic Logic 61 (1996) 705–744)?
Motivation / Meta: I am mostly trying to get an intuitive image of how intuitionistic ordinals work, and $\mathscr{C}_\alpha, \mathscr{S}_\alpha, \mathscr{N}_\alpha$ are not only objects I can (sort of) visualize but they also seem fairly natural, hence the question. I am a little confused by the fact that upper semicontinuous functions seem to be necessary to make $<$ well-founded (see above), while lower semicontinuous functions would have made the connection with $\Omega$ more transparent — I may have missed something stupid. If a closely related object is better behaved, I am interested in answers concerning it.
Edit (2019-03-21) / Meta: In a previous edit to this question (see revision history) I thought I had defined a sheaf $\mathscr{T}_\alpha$ of lower semicontinuous $\alpha$-valued functions which was better behaved (notably, both well-founded and $<$-extensional; with $\mathscr{T}_2$ being $\Omega$ in the more obvious way $V \mapsto \mathbf{1}_V$ and $\mathscr{T}_\omega$ being locally bounded lower semicontinuous $\mathbb{N}$-valued functions). It turns out my definition was nonsensical, so I'm deleting it. If I find a satisfactory definition, I'll edit again. But maybe such an object cannot be defined and that's part of the problem. Sorry about the noise.
