Characteristic Impedance without transmission lines
1) What is Z0? There are no transmission lines here.


The figure shows the power divider connected to three transmission lines. The parallel lines drawn extending away from the points marked \$V_1\$, \$V_2\$, and \$V_3\$ are representations of transmission lines. It's unfortunate for introductory material that they put the "Port" labelings at the far ends of the lines rather than next to the \$V_n\$ dots, because that is where the ports actually are — but they probably meant the labels merely to indicate the numbering of the ports on the actual divider.

For purposes of your reflection calculation, \$Z_0\$ is just “the impedance of the port attached to Port 1 of the power divider”. If what's attached is a transmission line, then the impedance is its characteristic impedance. You cannot calculate the reflection coefficient of a port without specifying the impedance attached to the port, because the amount of reflection is determined by the impedance match or mismatch.


  I thought that maybe Z0 is like a reference value, which can be used in any kind of network.


Sort of. Remember, an ideal transmission line can be of any length, including zero.  Therefore, in any model including a transmission line, we can reduce the length of the line to zero to model directly attached components. Thus, take the depicted situation, of being attached to a line with impedance \$Z_0\$, not as “there must be a transmission line here” but “there is something here that itself has a port with impedance \$Z_0\$, which is often a transmission line”. (The ends of transmission lines are ports — or more rigorously, if a transmission line is attached to a port then the other end of the transmission line is also a port.)

The practical reason to talk somewhat sloppily about \$Z_0\$ in this way is that we want system designs that can be used with transmission lines wherever we need them, so we choose a standard impedance for all of our ports and lines (except for those where we want reflections in some cases, such as filters), and call that \$Z_0\$, generalizing the concept of “characteristic impedance” to “the standard impedance we're using”. Further, we choose that standard impedance to be the characteristic impedance of the design of transmission line we would like to use.

You can end up using \$Z_0\$ as a reference value that no element of the circuit actually exhibits. But in this case, the diagram supposes that the divider is attached to transmission lines or other devices whose ports have impedance \$Z_0\$.