= Solution
For fixed $\beta$, the recursion defining $\beta+\alpha$ makes it continuous in $\alpha$ at every nonzero limit, and $(\beta+\alpha)+1>\beta+\alpha$. Thus \b[$\alpha\mapsto\beta+\alpha$ is normal], including when $\beta=0$.
For $\beta>0$, take $\lambda=\beta\cdot\omega$. If $\alpha<\lambda$, choose a <natural number> $n$ with $\alpha<\beta n$; then $\alpha+\beta\leq\beta(n+1)<\lambda$. Since $\alpha+\beta\geq\alpha$, their <supremum> as $\alpha\uparrow\lambda$ is $\lambda$. But $\lambda+\beta>\lambda$, so
$$
\sup_{\alpha<\lambda}(\alpha+\beta)=\lambda<\lambda+\beta.
$$
Hence right addition by a nonzero fixed <ordinal> fails continuity and is not normal.
<Normality and argument order in ordinal arithmetic> also give the multiplication statements: \b[$\alpha\mapsto\beta\cdot\alpha$ is normal for $\beta>0$, while $\alpha\mapsto\alpha\cdot\beta$ is normal exactly when $\beta=1$]. At $\beta=0$ it is constant. For fixed $\beta>1$, let $\kappa=\omega^{\beta\cdot\omega}$. <Cantor normal form> shows that $\xi\beta<\kappa$ whenever $\xi<\kappa$: leading exponents below $\beta\omega$ remain below it after adding the leading exponent of $\beta$. Thus $\sup_{\xi<\kappa}\xi\beta=\kappa$, but $\kappa\beta>\kappa$, again violating continuity. Left-fixed multiplication is strictly increasing and continuous by its recursion.
For exponentiation, \b[$\alpha\mapsto\beta^\alpha$ is normal for $\beta\geq2$; bases zero and one are exceptions. The reverse map $\alpha\mapsto\alpha^\beta$ is normal exactly when $\beta=1$]. Exponent zero gives the constant one. For $\beta>1$, use the same $\kappa$ and put $\lambda=\omega^\kappa$. For each $\alpha<\lambda$ choose $0<\zeta<\kappa$ with $\alpha\leq\omega^\zeta$. Then $\alpha^\beta\leq\omega^{\zeta\beta}<\lambda$, while $\alpha^\beta\geq\alpha$ for positive $\alpha$. Consequently its left <supremum> is $\lambda$, but $\lambda^\beta=\omega^{\kappa\beta}>\lambda$. The bound by $\omega^\zeta$ also handles finite bases, where an unqualified leading-exponent multiplication formula would need care. All these are <ordinal> facts and use no choice.
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