Let $f: [0,\infty) \to [0,\infty)$ and $s,t \in [0,\infty)$. Then
$$\DeclareMathOperator{cov}{cov} \cov(B_{f(s)},B_{f(t)}) = \min \{f(s),f(t)\}$$
since $\cov(B_u,B_v)=\min\{u,v\}$ holds for all $u,v \in [0,\infty)$, so in particular for $u:=f(s)$, $v:=f(t)$.
Concerning your second question:
Theorem: Let $(X_t)_{t \geq 0}$ a continuous local martingale, $X_0=0$. Then
there exists a Brownian motion $(W_t)_t$ (with respect to a filtration
$(\mathcal{G}_t)_{t \geq 0}$) such that $X_t = W_{[X]_t}$ where $[X]$
denotes the compensator of $X$.
(You can find some more information here, it's theorem 1.)
In this case: If there exists a function $g$ such that $f(t)=\int_0^t g(s)^2 \, ds$, then we can define
$$X_t := \int_0^t g(s) \, dB_s$$
which gives $[X]_t = \int_0^t g(s)^2 \, ds=f(t)$. By applying the theorem we obtain that there exists a Brownian motion $(W_t)_{t \geq 0}$ such that $$X_t = \int_0^t g(s)^2 \, dB_s = W_{[X]_t} = W_{f(t)}$$
This works in particular if $f$ is differentiable and $f(0)=0$, put $g(s) := \sqrt{f'(s)}$. Of course that's not what you were looking for since you wanted to find a representation as an Itô process for a given Brownian motion.
The problem is the following: You could define (as above)
$$X_t := \int_0^t g(s) \, dB_s$$
where $g(s) := \sqrt{f'(s)}$, $f(0)=0$. Then $X_t$ is a centered Gaussian random variable with variance $f(t)$ and $X_t-X_s$ has variance $f(t)-f(s)$ which means that $X_t$ has the same distribution as the time-changed Brownian motion $B_{f(t)}$. But they are not necessarily the same processes! If you choose for example $f(t):=2t$ you obtain
$$X_t = \int_0^t \sqrt{2} \, dB_s = \sqrt{2} B_t$$
and this is clearly not the same process as $B_{f(t)} = B_{2t}$. On the other hand we know that $W_t := \sqrt{2} \cdot B_{\frac{t}{2}}$ is a Brownian motion and obviously $$X_t = \sqrt{2} \cdot B_{t} = \sqrt{2} \cdot B_{\frac{2t}{2}} = W_{2t} = W_{f(t)}$$
Remark:
- Let $X_t := \int_0^t f_s \, dB_s$ where $f: \Omega \times \mathbb{R} \to \mathbb{R}$ such that $\mathbb{E}\left(\int_0^t f(s)^2 \, ds\right)<\infty$. Then $(X_t)_t$ is a martingale and the compensator is given by $$[X]_t = \int_0^t f(s)^2 \, ds$$ If $f$ is deterministic, i.e. $f: \mathbb{R} \to \mathbb{R}$, we know therefore that $(X_t)$ has a deterministic compensator.
- Let's consider the process $(t,\omega) \mapsto B(f(t),\omega)$ where $f: [0,\infty) \to [0,\infty)$ is a deterministic injective function such that $f(0)=0$. In this case $f$ determines how fast you run through the trajectories $(t,\omega) \mapsto B(t,\omega)$. For example $f(t):=2t$ means that you run through the trajectories twice as fast.
For #1: Let's call your isomorphism $T$, the isometry from $\mathcal{H}_K$ to $L^2(P)$ which maps $K_t$ to $B_t$. Yes, $T$ really is a stochastic integral: $Tf$ is the Itô integral, not of $f$, but of $f'$. So
$$Tf = \int_0^1 f'(t)\,dB_t$$
or in other words, for deterministic $g \in L^2([0,1])$,
$$\int_0^1 g(t)\,dB_t = T\left(\int_0^\cdot g(s)\,ds\right).$$
So this recovers the Itô integral for deterministic $L^2$ integrands. This special case of the Itô integral is sometimes called the Wiener integral.
Another way of thinking about this is that the map $f \mapsto f'$ is an isometric isomorphism from $\mathcal{H}_K$ to $L^2([0,1])$. So if you identify $\mathcal{H}_K$ with $L^2([0,1])$ under this map, then $T$ really is the stochastic integral.
I will think about your other questions some more.
Best Answer
So that this has a (non-deleted) answer:
No such process can exist. If $\operatorname{Var}(W_0) = \max(0,0) = 0$ then $W_0$ is a constant. In particular, $W_0$ and $W_1$ are independent, so we have $\operatorname{Cov}(W_0, W_1)= 0 \ne 1= \max(0,1)$.
More generally, Cauchy-Schwarz shows that for any process with finite variance, we have $\operatorname{Cov}(X_s, X_t) \le \sqrt{\operatorname{Var}(X_s) \operatorname{Var}(X_t)}$. Thus the covariance function $k(s,t)$ must satisfy $k(s,t) \le \sqrt{k(s,s) k(t,t)}$. The function $k(s,t) = \max(s,t)$ does not, so it cannot be a covariance function.