The answer to Question 3 is yes, and to Questions 1,2 the answer is "no", unless the initial conditions are very special.

To address these questions, assume wlog that $T=2\pi$, and expand everything into Fourier series. Let $a_n, b_n$ be coefficients of $r_1,r_2$, and $s_n$ coefficients of $s$. Let $D_i=P_i(D)$, where $D=d/dx$. Let $c_{i,n}=P_i(n\sqrt{-1})$.

I assume that the quantity minimized is the sum of the squares of $L^2$ norms over $(0,T)$, though the notation of Alexander is strange.

Let us address Question 3 first (no initial conditions). Then a simple computation shows that there is a minimum and its Fourier coefficients are $$s_n=\frac{a_nc_{1,n}+b_nc_{2,n}}{c_{1,n}^2+c_{2,n}^2}.$$ (Substitute the Fourier series, use Parseval, and solve the simple quadratic minimization problem). This solution is unique if $P_i$ have no common root (so the denominator is never $0$). If they have a common root, an exponential polynomial can be added to $s$ without affecting the minimum.

This gives a periodic solution. Well, we solved the problem on the interval $(0,T)$ and ANY function can be extended periodically. But one wants the extention to be smooth enough, so that operators $D_i$ make sense. The solution formula shows that the solution is in general has roughly $1$ more derivative in comparison with $r_i$. So if the $r_i$ are smooth enough than $s$ is smooth and periodic with the same period.

If we solve the same problem (without initial conditions) on an interval $(0,mT)$, but the $r_i$ still have period $T$, then the formula shows that the solution is the same (has period $T$) unless we are in the unlikely situation when the $P_i$ have a common root. But even in this situation, there always exists a solution with period $T$.

Now what happens when we add the initial condition (Questions 1,2). These conditions add a linear restriction on the $s_j$, so when we optimize on $(0,mT)$, in general harmonics $\exp(in/m)$ will be present.

And of course their contribution cannot tend to $0$ when $x\to\infty$ because they are periodic.

The answer to Question 3 is yes, and to Questions 1,2 the answer is "no", unless the initial conditions are very special.

To address these questions, assume wlog that $T=2\pi$, and expand everything into Fourier series. Let $a_n, b_n$ be coefficients of $r_1,r_2$, and $s_n$ coefficients of $s$. Let $D_i=P_i(D)$, where $D=d/dx$. Let $c_{i,n}=P_i(n\sqrt{-1})$.

The quantity minimized is the sum of the squares of $L^2$ norms over $(0,T)$.

Let us address Question 3 first (no initial conditions). Then a simple computation shows that there is a minimum and its Fourier coefficients are $$s_n=\frac{a_nc_{1,n}+b_nc_{2,n}}{c_{1,n}^2+c_{2,n}^2}.$$ (Substitute the Fourier series, use Parseval, and solve the simple quadratic minimization problem). This solution is unique if $P_i$ have no common root (so the denominator is never $0$). If they have a common root, an exponential polynomial can be added to $s$ without affecting the minimum.

This gives a periodic solution. Well, we solved the problem on the interval $(0,T)$ and ANY function can be extended periodically. But one wants the extention to be smooth enough, so that operators $D_i$ make sense. The solution formula shows that the solution is in general has roughly $1$ more derivative in comparison with $r_i$. So if the $r_i$ are smooth enough than $s$ is smooth and periodic with the same period.

If we solve the same problem (without initial conditions) on an interval $(0,mT)$, but the $r_i$ still have period $T$, then the formula shows that the solution is the same (has period $T$) unless we are in the unlikely situation when the $P_i$ have a common root. But even in this situation, there always exists a solution with period $T$.

Now what happens when we add the initial condition (Questions 1,2). These conditions add a linear restriction on the $s_j$, so when we optimize on $(0,mT)$, in general harmonics $\exp(in/m)$ will be present.

And of course their contribution cannot tend to $0$ when $x\to\infty$ because they are periodic.