First, a note regarding proper classes: One can construct a tuple of proper classes, for example with the Morse definition. Same goes for proper-class-sized algebraic structures, as in the field of surreal numbers. One can construct the class of subclasses, or power class, of a proper class. Finally, one can construct proper-class-sized relations and therefore proper-class-sized functional relations, which I'll just refer to as functions.
Let $\text{Ord} = \mathbb{N}_\text{Ord}$ be the class of ordinals. Let $+$ and $\times$ be the Hessenberg sum and product, respectively. Let $<$ be the usual order (set membership). Then $(\mathbb{N}_\text{Ord}, <, +, \times)$ is an ordered rig. Let $\mathbb{Z}_\text{Ord} = (\mathbb{N}_\text{Ord} \times \mathbb{N}_\text{Ord}) / \sim_\mathbb{Z}$ where \begin{align} (a_1, a_2) \sim_\mathbb{Z} (b_1, b_2) &\leftrightarrow a_1 + b_2 = a_2 + b_1 \\ [(a_1, a_2)] < [(b_1, b_2)] &\leftrightarrow a_1 + b_2 < a_2 + b_1 \\ [(a_1, a_2)] + [(b_1, b_2)] &= [(a_1 + b_1, a_2 + b_2)] \\ [(a_1, a_2)] [(b_1, b_2)] &= [(a_1 b_1 + a_2 b_2, a_1 b_2 + a_2 b_1)] \end{align}
Then $(\mathbb{Z}_\text{Ord}, <, +, \times)$ is an ordered ring. Let $\mathbb{Q}_\text{Ord} = (\mathbb{Z}_\text{Ord} \times \mathbb{Z}_\text{Ord} {\setminus} \{0\}) / \sim_\mathbb{Q}$ where \begin{align} (a_1, a_2) \sim_\mathbb{Q} (b_1, b_2) &\leftrightarrow a_1 b_2 = a_2 b_1 \\ [(a_1, a_2)] < [(b_1, b_2)] &\leftrightarrow \begin{cases} a_1 b_2 < a_2 b_1 & a_2 b_2 > 0 \\ a_1 b_2 < a_2 b_1 & a_2 b_2 < 0 \\ \end{cases} \\ [(a_1, a_2)] + [(b_1, b_2)] &= [(a_1 b_2 + a_2 b_1, a_2 b_2)] \\ [(a_1, a_2)] [(b_1, b_2)] &= [(a_1 b_1, a_2, b_2)] \end{align}
Then $(\mathbb{Q}_\text{Ord}, <, +, \times)$ is an ordered field. How can we extend this construction to a "complete" ordered field $\mathbb{R}_\text{Ord}$? In the direction of Dedekind cuts, we might have $\mathbb{R}_\text{Ord} \subseteq \mathcal{P}(\mathbb{Q}_\text{Ord})$ be the class of nonempty, proper, downward closed subclasses $a$ without a greatest element: \begin{align} &a \neq \{\} \\ &a \neq \mathbb{Q}_\text{Ord} \\ &\forall x: \forall y: (x < y \in a) \rightarrow x \in a \\ &\forall x: (x \in a) \rightarrow \exists y: (x < y \in a) \end{align}
with \begin{align} a < b &\leftrightarrow a \subset b \\ a + b &= \{x + y : x \in a, y \in b\} \end{align}
and a more complex formula for the product like that for Dedekind cuts on $\mathbb{Q}$. In the direction of Cauchy sequences, we might have $\mathbb{R}_\text{Ord} = X / \sim$ where $X \subseteq \mathbb{N}_\text{Ord} \rightarrow \mathbb{Q}_\text{Ord}$ is the class of convergent transfinite sequences $a$: \begin{align} \forall \varepsilon \in \mathbb{Q}_\text{Ord}: \varepsilon > 0 \rightarrow \exists n \in \mathbb{N}_\text{Ord}: \forall i, j \in \mathbb{N}_\text{Ord}: i, j > n \rightarrow |a(i) - a(j)| < \varepsilon \end{align}
with \begin{align} a \sim b &\leftrightarrow \forall \varepsilon \in \mathbb{Q}_\text{Ord}: \varepsilon > 0 \rightarrow \exists n \in \mathbb{N}_\text{Ord}: \forall i \in \mathbb{N}_\text{Ord}: i > n \rightarrow |a(i) - b(i)| < \varepsilon \\ [a] < [b] &\leftrightarrow \exists n \in \mathbb{N}_\text{Ord}: \forall i \in \mathbb{N}_\text{Ord}: i > n \rightarrow a(i) < b(i) \\ [a]+[b] &= [n \mapsto a(n) + b(n)] \\ [a][b] &= [n \mapsto a(n) b(n)] \end{align}
Are these two directions valid (e.g. are they class-theoretically sound, ordered fields, and "complete" in a reasonable sense)? Are they equivalent or isomorphic? What is their relationship with the field of surreal numbers?