Analogue filter analysis (band-pass)
Even though this may not be homework, it is a basic textbook problem, and for this reason I just give you a few hints, which should nevertheless help you solve the problem if you put in a bit of effort.


All questions refer to the filter's frequency response, which is simply obtained by evaluating $G(s)$ for $s=j\omega$.
The first question should read "Show that this property ... $\omega=0$, $\omega={\infty}$" (not $\omega=1$).
Your expression for $|G(j\omega)|$ is wrong; you wrote down $G(j\omega_0)$ (without magnitude). Of course, if $G(j\omega)=0$, or $G(j\omega)=1$, then also $|G(j\omega)|=0$ and $|G(j\omega)|=1$, respectively.
In order to see what happens for $\omega\rightarrow\infty$, write down $|G(j\omega)|^2$ (that's maybe easier than the magnitude), and take the limit.
For the 3 dB cut-off frequencies (there must be two positive solutions), use $|G(j\omega)|^2$ derived above and solve $|G(j\omega_c)|^2=\frac12$. Substitute $x=\omega^2$ and solve the resulting quadratic equation for $x$. From that solution, derive the two positive solutions for $\omega_c$.


All of this involves basic math skills, such as computing the (squared) magnitude of a complex number, and solving quadratic equations. If you find that difficult, read up on these basics because you'll need them all of the time when doing signal processing.