For all ring with unit element $A$ let $W(A)$ be the ring of $p$-typical Witt vectors. Denote by $$\phi\;:\;W(A)\to A^{\mathbb{N}}$$ the ghost map, which is given by $$\phi(a_0,a_1,a_2,\ldots)\;=\;(\phi_0,\phi_1,\phi_2,\ldots)$$ where $\phi_n=\phi_n(a_0,\ldots,a_n)$ is defined by $$\phi_n=a_0^{p^n}+pa_1^{p^{n-1}}+p^2a_2^{p^{n-1}}+\cdots+p^na_n\;.$$

I have 3 questions. The first is general, while the second and the third seem more approachable.

ASSUMPTION. $A$ is the ring of integers of a complete ultrametric field $K$ containing the field of $p$-adic numbers $\mathbb{Q}_p$. Denote by $k$ the residual field of $A$, it is a field of positive characteristic $p$.

QUESTION 1: *Under the above assumption, I'm interested in understanding the image of the map $\phi$.*

Notice that the map $\phi:W(K)\to K^{\mathbb{N}}$ is bijective.

Allow me to add some useful information. Under the assumption, $\phi:W(A)\to A^{\mathbb{N}}$ is injective and it is also known that **if** there is a ring endomorphism $\sigma:A\to A$ such that $\sigma(a)\equiv a^p \pmod{pA}$ for all $a\in A$, then a sequence $(\phi_0,\phi_1,\phi_2,\ldots)$ lies in $\operatorname{Im}(\phi)$ if and only if for all $n\geq 0$ one has $$\sigma(\phi_n)\equiv\phi_{n+1}\pmod{p^{n+1}A}$$
(for a proof see Bourbaki Alg. Comm. Chapter 9, $\S1$, $N.2$, Prop. 2, page AC IX.3). This criterion is due to B.Dwork.

The criterion provides an answer to Question 1 when $K$ is unramified over $\mathbb{Q}_p$ (i.e. the maximal ideal of $A$ is $pA$), because in this case $A=W(k)$ has a canonical Frobenius. What's going on in the general case? As soon as the ideal $pA$ is not maximal (ramification) the presence of a Frobenius endomorphism of $A$ (i.e. a ring endomorphism of $A$ lifting of the $p$-th power of $k$) seems not enough to guarantee a description of the image by means of congruences as above.

I also have the following important related questions, that might be more approachable.

QUESTION 2: *Maintain the above assumption. Can we describe the set of elements $a\in A$ such that there exists $x=(x_0,x_1,\dotsc)\in W(A)$ with $$\phi(x_0,x_1,\dotsc)=(a,a,a,\dotsc)\quad ?$$*
In other words, what is the set of $a\in A$ such that we can solve in $A$ the system of conditions
\begin{align*}
x_0&{}=a \\
x_0^p+px_1&{}=a \\
x_0^{p^2}+px_1^p+p^2x_2&{}=a \\
\cdots&\qquad?
\end{align*}
Of course, if $a$ lies in $\mathbb{Z}_p=W(\mathbb{F}_p)$ the ring of $p$-adic integers, then $a$ is a solution of my problem (by the above congruences criterion). Moreover, if $A=W(k)$, then Dwork's criterion proves that the only solutions $a\in A$ are the elements of $\mathbb{Z}_p$.

In general, the set of such elements $a$ is a sub-ring $B$ of $A$ containing $\mathbb{Z}_p$.

Question 2 is relevant for some convergence properties of certain Artin–Hasse exponentials that I do not mention here. However, it is also related to the following interesting question.

QUESTION 3: *The ring $A^{\mathbb{N}}$ is naturally an $A$-algebra via the diagonal map $A\to A^{\mathbb{N}}$ sending $a\mapsto (a,a,a,\dotsc)$. Now, when does this $A$-algebra structure induce an $A$-algebra structure on $W(A)$ such that $\phi:W(A)\to A^{\mathbb{N}}$ is $A$-linear?*

Alternatively, can we describe the maximal sub-ring $B\subseteq A$ such that $W(A)$ has a $B$-module structure making $\phi:W(A)\to A^{\mathbb{N}}$ a $B$-linear ring homomorphism?

Any comment will be useful and really appreciated.