$\mathbb{Z}_p^{\mathbb{N}}$-extension and formal Drinfeld module

From the notation of the paper, $A = \mathbb{F}[\theta]$ and $F = \mathbb{F}(\theta)$, for a finite field $\mathbb{F}$ and a variable $\theta$. They let $\Phi : A \to A\{\tau\}$ be the Carlitz module defined via $\Phi_{\theta} = \theta + \tau$ and extended uniquely so that $\Phi$ is an $\mathbb{F}$-algebra homomorphism.

Take $\mathfrak{p}$ to be a maximal ideal of $A$, and let $\pi_{\mathfrak{p}} \in A$ be a monic generator of $\mathfrak{p}$. For $n \geq 0$, let $F_n = F(\Phi[\mathfrak{p}^{n+1}])$, where $$\Phi[\mathfrak{p}^{n+1}] = \Phi[\pi_{\mathfrak{p}}^{n+1}] = \{ x \in \overline{F} : \Phi_{\pi_{\mathfrak{p}}^{n+1}} = 0 \}$$ is the $\mathfrak{p}^{n+1}$-torsion of the Carlitz module, which is isomorphic to $A/\mathfrak{p}^{n+1}$ as an $A$-module. There is an old result of Carlitz that says that there is a group isomorphism $\kappa : \mathrm{Gal}(F_n/F) \stackrel{\sim}{\to} (A/\mathfrak{p}^{n+1})^{\times}$ such that for $\varepsilon \in \Phi[\mathfrak{p}^{n+1}]$ and $\sigma \in \mathrm{Gal}(F_n/F)$, $$\sigma(\varepsilon) = \Phi_{\kappa(\sigma)}(\varepsilon). \tag{1}$$ Strictly speaking $\kappa(\sigma)$ is not an element of $A$, but we can take $\Phi_{\kappa(\sigma)} \in A\{ \tau\}$ for any lift of $\kappa(\sigma)$ in $A$ modulo $\mathfrak{p}^{n+1}$, and it is independent of the choice of lift. For more details one can read section 7.1 of Drinfeld modules by Papikian.

Everything the authors do next is part of passing to the limit as $n \to \infty$. They let $\mathcal{F} = \cup_n F_n = F(\Phi[\mathfrak{p}^{\infty}])$, in which case we can extend $\kappa$ to an isomorphism $$\kappa: \mathrm{Gal}(\mathcal{F}/F) \xrightarrow{\sim} A_{\mathfrak{p}}^{\times} \quad (\cong \varprojlim (A/\mathfrak{p}^{n+1})^{\times}),$$ where $A_{\mathfrak{p}}$ is the completion of $A$ at $\mathfrak{p}$. In order to extend the identity in (1), the authors need to extend the definition of the Carlitz module to a formal Drinfeld module $\Phi : A_{\mathfrak{p}} \to A_{\mathfrak{p}}\{ \tau\}$ (e.g., see section 6.5 of Papikian). Once this is done, the equation in (1) still holds, but with $\Phi$ now replaced with its formal $A_{\mathfrak{p}}$-module, $\kappa$ replaced with its extension to $\mathcal{F}$, and $\varepsilon$ allowed to be any element of $\Phi[\mathfrak{p}^{\infty}]$. Because all of these extensions arise naturally the authors abuse notation only slightly by using the $\Phi$ and $\kappa$ for technically two different things. I think this addresses your first two questions.

For the third question what goes awry is that there is no $\mathfrak{p}$-adic logarithm for $A_{\mathfrak{p}}^{\times}$, since one would need to divide by the characteristic to define it. On the other hand, you might consult Proposition 1.6 in Number Theory in Function Fields by Rosen, where he describes why the number of necessary generators for $(A/\mathfrak{p}^n)^{\times}$ goes to infinity. A more detailed analysis can be found in Lemma 4.2 of

C. Li and J. Zhao, "Class number growth in a family of $\mathbb{Z}_p$-extensions over global function fields," J. Algebra 200 (1998), 141-154.