What is the general form of a first and second order dynamic system, in laplace and time space?
You want to use the following property of Laplace transform:
$$\mathscr{L}\left({\frac{dx(t)}{dt}}\right)(s) = s\mathscr{L}\left(x(t)\right)-x(0)$$

This allows you to easily move between differential equations and polynomial equations.

Time domain to frequency domain: Take your first equation for example
$$\frac{d^2 x(t)}{dt^2} + 2 \zeta \omega_n \frac{dx(t)}{dt} + \omega_n^2 x(t) = f(t).$$

If we denote the Laplace transform of \$x(t)\$ by \$X(s)\$ and \$f(t)\$ by \$F(s)\$ and apply Laplace transform to this equation then this property implies
$$s^2 X(s) + 2 \zeta \omega_n sX(s) + \omega_n^2 X(s) = F(s)$$
where for simplicity I assume that \$x(0) = f(0) = 0\$.

The transfer function is defined as the ratio:
$$\frac{X(s)}{F(s)} = \frac{1}{s^2+2 \zeta \omega_n s + \omega_n^2}$$

Frequency domain to time domain: Lets try the example \$G(s)=\frac{1}{s^3+1}\$ then we have by definition that
$$G(s)F(s) = X(s)$$
which implies
$$F(s) = s^3X(s)+ X(s).$$

Taking inverse Laplace transform we find that
$$f(t) = \frac{d^3 x(t)}{dt^3} + x(t).$$

Hopefully this allows you to see the pattern in general.