Consider a system of differential equations
$$ \frac{d}{dt}f(t) = F(t, f(t), g(t)), $$ $$ \frac{d}{dt}g(t) = G(t, f(t), g(t)). $$
Assume $F, G \in C^{\infty}$. What is the necessary and sufficient condition on which the solution of this system approaches a unique limiting point (can be fixed or periodic or any other limiting point which might depend on time) no matter what the initial condition $f(0)$ and $g(0)$ is?
If there is a unique solution $(f, g)$ for
$$ 0 = F(t, f, g), $$ $$ 0 = G(t, f, g). $$ Can we say that $\lim_{t\rightarrow \infty} f(t) = f$ and $\lim_{t\rightarrow \infty} g(t) = g$?
The answer to your second question is no : let $ F(t,f,g) = g$ and $G(t,f,g) = -f$. Then, the only solution to $F=G= 0$ is $f=g= 0$, but the solution to the Cauchy problem is given by : \begin{align} f(t) &= \cos(t)f (0) + \sin(t) g(0) \\ g(t) &= - \sin(t)f(0) + \cos(t)g(0) \end{align}
As for your first question, I think the problem is too broad for there to be a real answer.