Let $A$ be the Kronecker algebra over an algebraically closed field. I know that the indecomposable $A$-modules fall into the preinjective component $\mathcal{I}$, the postprojective component $\mathcal{P}$ and the regular components $\mathcal{R}_\lambda$ for $\lambda$ on the projective line.
Now, it seems that there are short exact sequences $$0 \to P_n \to P_m \to R_{t, \lambda} \to 0 $$ for $m>n$ and $t=m-n$. (We denote by $P_n$ the $n$-th indecomposable postprojective $A$-module, i.e. the one with dimension $2n+1$.) However, I have only been able to construct such sequences for $m-n=1$. I've done so explicitly with the help of linear algebra, but don't know how to generalize my construction.
My questions is: Do these sequences really exist for any $m>n$ and is there a more practical/general approach than what I tried to do?
It's easy to construct such sequences when $n=0$, so that the left hand term is simple projective.
I will do an example for $m=3$, which generalises to all $m$. Take any monic polynomial $f=t^3+at^2+bt+c\in K[t]$. Here $K$ is any field, not necessarily algebraically closed. We use the coeffients to form a map $(c,b,a,1)^t\colon K\to K^4$. The cokernel $K^{m+1}\to K^m$ is then represented by the matrix $$ \begin{bmatrix}1&0&0&-c\\0&1&0&-b\\0&0&1&-a\end{bmatrix}. $$ Now the Kronecker module $P_3$ can be represented by the pair of matrices $(A,B)\colon K^3\to K^4$, where $$ A = \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\\0&0&0\end{bmatrix} \quad\textrm{and}\quad B = \begin{bmatrix}0&0&0\\1&0&0\\0&1&0\\0&0&1\end{bmatrix}. $$ We have the map $(c,b,a,1)^t\colon P_0\to P_4$, and the cokernel is thus the Kronecker module $R_f$, represented by the pair of matrices $(I,C(f))\colon K^3\to K^3$, where $I$ is the identity and $C(f)$ is the companion matrix to $f$, so $$ C(f) = \begin{bmatrix}0&0&-c\\1&0&-b\\0&1&-a\end{bmatrix}. $$
Now using standard properties of companion matrices, if $f=f_1^{s_1}\cdots f_r^{s_r}$ is the factorisation into monic irreducible polynomials, then the companion matrix is similar to the direct sum $C(f_1^{s_1})\oplus\cdots\oplus C(f_r^{s_r})$, so the module $R_f$ is isomorphic to the direct sum $R_{f_1^{s_a}}\oplus\cdots\oplus R_{f_r^{s_r}}$. Moreover, each $R_{f_i^{s_i}}$ is indecomposable regular, of regular-length $s_i$ and regular-socle $R_{f_i}$.
If $K$ is algebraically closed, then $f_i=t-\lambda_i$, and we recover your $R_{s_i,\lambda_i}$, but as you see, the theory goes through for general fields $K$: the regular simples are indexed by the closed points of the scheme $\mathbb P^1_K$, or homogeneous prime ideals of $K[t,u]$, or prime ideals of $K[t]$ together with $\infty$.
OK, so that deals with the case when $n=0$. In general, there are some nice endofunctors of the category of Kronecker modules which allows one to shift everything to the left or right. The pushout functor takes a Kronecker module $(A,B)\colon U\to V$, forms the pushout diagram $$ \require{AMScd} \begin{CD} U @>A>> V\\ @VBVV @VVDV\\ V @>>C> W \end{CD} $$ and returns the Kronecker module $(C,D)\colon V\to W$.
This functor is right exact, sends $P_m$ to $P_{m+1}$, and sends the regular $R_f$ to itself. (Similarly we have the pullback functor, and these form an adjoint pair.)
Checking dimensions, we see that it sends our short exact sequence $$ 0 \to P_0 \to P_m \to R_f \to 0 $$ ($f\in K[t]$ monic of degree $m$) to a short exact sequence $$ 0 \to P_1 \to P_{m+1} \to R_f \to 0. $$ Iterating, we obtain short exact sequences $$ 0 \to P_n \to P_{m+n} \to R_f \to 0 $$ for all $n\geq0$.