Is it possible to apply a pseudo-random permutation (a keyed permutation) to construct a sponge function?
Yes, it is possible to construct a keyed-sponge function $\mathcal{F}_k(x, n)$ using a pseudo-random permutation $\pi$ and a secret key $k$. Here, $x$ refers to the input message, and $n$ to the number of output bits.
The sponge construction consists of $\pi$, and a state $S$, which is $s$-bits (sometimes also denoted $b$-bits). $S$ is conceptually divided into two parts: the leading section, called the rate $R$, which is $r$-bits; and, the trailing section, called the capacity $C$, which is $c$-bits. The following equation holds: $b = s = r + c$. For how $x$ is absorbed into $R$, see this.
I'll use $k_{enc}$ here as shorthand to refer to $k_{enc} = \textrm{encode_key(}k, \mathcal{F}\textrm{)}$ to abstract away some of the complexity involved in domain-separation and how $k$ is preprocessed to fit within a certain portion of $S$. The function $\textrm{encode_key(}k, \mathcal{F}\textrm{)}$ may encode $k$ with concatenated metadata, like its length, and pad it with a function-specific value to the appropriate size. The exact metadata, value, size and portion of $S$ may be defined in different ways by the designer of $\mathcal{F}$ and the specific approach used in the construction.
There are several approaches for how to construct a keyed-sponge mentioned in the literature. There's the Outer-Keyed Sponge (OKS), which absorbs $k_{enc}$ into $R$ prior to absorbing $x$ into $R$. Also, the Inner-Keyed Sponge (IKS), which sets $C = k_{enc}$ prior to absorbing $x$ into $R$. And, there's the Full-Keyed Sponge (FKS), which sets $S = k_{enc}$ prior to absorbing $x$ into the full $S$. The KMAC construction, for instance, uses OKS to initialize its internal keyed-permutation. At a minimum, they all assume $k$ and $C$ will be kept secret, as well as all of the initial $S$ before the first call to $\pi(S)$ after $k_{enc}$ has been incorporated.
But the article “Sponge function” does not mention any presence of the key in the used permutation function $f(x)$. Does this imply that we can fix any public key $K$ and always use $f_K(x)$ as $f(x)$?
No, $k$ is expected to be kept secret to the communicating parties. Using a public key(1), or parameter, $K$ as the key for a keyed-sponge permutation is not going to give you the security properties you may be looking for.(2)(3)(4) Though, there is utility to including a public parameter $K$ in the input as something other than a key, like as a nonce, or for domain separation. In these cases, the length of $K$ may only incidentally affect the security provided by the nonce or domain-separator, insofar as they both need to be canonically unique to each usage context.
If no, then [...] how will the length of $k$ impact the security of the $n$-bit output?
In the paper Key Prediction Security of Keyed Sponges(5), the simplified PRF security bounds of the different approaches is defined by a notion called key-prediction security (key-pre). FKS is considered a generalization of IKS in this paper, so the same bounds apply to both. The output size $n$ is not directly considered in these bounds. In general, for sponges, the expected resistance against output collisions is ~$\min{ \left\{2^{n/2}, 2^{c/2}\right\} }$, and the expected resistance against inner collisions is ~$2^{c/2}$.(6 §6.2) The provided equations to estimate aversarial advantage refer to: $M$, the number of queries an adversary makes to $\mathcal{F}_{\bar k}(\bar{x}, n)$; $N$, the number of queries an adversary makes to $\pi{(\bar{S})}$; and, $\lambda$, which applies to OKS, and is the number of rounds of $\pi(S)$ that are performed when absorbing $k_{enc}$ into $R$. $\bar{k}$, $\bar{x}$ and $\bar{S}$ are guesses the adversary makes for the values $k$, $x$ and $S$, respectively.
$\boldsymbol{ \mathrm{Adv}_{\mathcal{F}}^{ \mathrm{key-pre} } }(N)$ is the probability that the adversary made queries to the underlying permutation that match the key input to $\mathcal{F} \in$ $\{$OKS, IKS, FKS$\}$ $-§1.1$
\begin{equation}
\label{eq:key-prediction-security-oks}
\boldsymbol{ \mathrm{Adv}_{\boldsymbol{ \mathrm{OKS} }}^{ \mathrm{key-pre} } }(N) \lesssim c^{\lambda - 1} \frac{N}{2^{|k|}}
\tag{Ref. a}
\end{equation}
\ref{eq:key-prediction-security-oks}: §1.2, Equation 4
\begin{equation}
\label{eq:key-prediction-security-fks}
\boldsymbol{ \mathrm{Adv}_{\boldsymbol{ \mathrm{FKS} }}^{ \mathrm{key-pre} } }(N) \le \frac{N}{2^{|k|}}
\tag{Ref. b}
\end{equation}
\ref{eq:key-prediction-security-fks}: §1.2, Equation 2
\begin{equation}
\label{eq:prf-security}
\boldsymbol{ \mathrm{Adv}_{\mathcal{F}}^{ \mathrm{prf} } }(M, N) \approx \frac{M^2}{2^c} + \frac{MN}{2^c} + \boldsymbol{ \mathrm{Adv}_{\mathcal{F}}^{ \mathrm{key-pre} } }(N)
\tag{Ref. c}
\end{equation}
\ref{eq:prf-security}: §1.1, Equation 1
And, when one also considers absorbing a nonce (which is used only once for a given $k$) into $R$ (or into all of $S$ for FKS) prior to absorbing $x$, the security bounds are shown to get a bit better:
\begin{equation}
\label{eq:prf-security-nonce-respecting}
\boldsymbol{ \mathrm{Adv}_{\mathcal{F}_{_{nonce}}}^{ \mathrm{prf} } }(M, N) \approx \frac{M^2}{2^s} + \frac{MN}{2^c} + \boldsymbol{ \mathrm{Adv}_{\mathcal{F}}^{ \mathrm{key-pre} } }(N)
\tag{Ref. d}
\end{equation}
\ref{eq:prf-security-nonce-respecting}: §1.3, Equation 5
